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Explore 2d Shapes Worksheets

Welcome to the Math Salamanders Explore 2d Shapes worksheets for kids. Here you will find a range of printable geometry worksheets, which will help your child to learn the properties of 2d shapes and develop their geometric reasoning skills.

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Explore 2d Shapes

The worksheets you find on this page have all been developed to try to help children get a better understanding of what 2d shapes are all about.

In particular, many children still find it hard to realize that a square is also a rectangle which is also a parallelogram which is also a quadrilateral which is also a polygon.

So one of the main development points of these sheets is to get children to know their shapes really well, and to understand what properties they have.

The other aim of these sheets is to try to get children to prove or demonstrate why a particular statement might be always true, sometimes true or never true in the context of geometry.

Reasoning is a key element in mathematics which is often neglected, and really gets children to think about their beliefs and why they think what they do.

Using these sheets will help your child to:

• know the properties of a range of 2d shapes;
• recognise that some shapes can also be described as being other shapes; e.g. a square is also a rhombus;
• recognise and understand right angles, parallel lines, lines of symmetry;
• develop their geometric reasoning skills.

Can You Draw It? Worksheets

The worksheets here involve children applying their knowledge and understanding to explore 2d shapes.

They are a great way to stretch the more able learners, or get children to think hard about the properties shapes have.

Although the sheets in this section are aimed at 3rd grade, they could easily be used by older children.

2d Shapes - Can you draw it?

• 2d Shapes - Can you draw it 1
• PDF version
• 2d Shapes - Can you draw it 2

Explore 2D Shapes Worksheets

All the sheets in this section involve saying whether a given statement is Always, Sometimes, or Never True.

There is also a section for children to explain the reasoning behind their thinking for each answer.

The worksheets have been split up into grades, starting with 3rd grade.

The grades provide a rough guide to where the children have developed enough skills to answer the questions.

However, you can use the worksheets with older children or even adults, to get them to develop their reasoning skills and explore 2d shapes.

With some sheets, there is a UK version of the sheet because of the difference in terminology of the word 'trapezium' and 'trapezoid'.

All of the other sheets can be used by anyone!

• Exploring 2d Shapes Sheet 1
• Exploring 2d Shapes Sheet 2
• Exploring 2d Shapes Sheet 3
• Exploring 2d Shapes Sheet 4
• Exploring 2d Shapes Sheet 4 UK version
• UK version Answers
• Exploring 2d Shapes Sheet 5
• Exploring 2d Shapes Sheet 5 UK Version
• UK Version Answers
• Exploring 2d Shapes Sheet 6
• Exploring 2d Shapes Sheet 7

Other Geometry Worksheets

• More 2d Shapes Worksheets
• Looking for some printable 2d shape worksheets?
• Need some symmetry worksheets?
• Looking for sheets about the properties of 2d shapes?

We have a huge selection of printable 2d shape worksheets for a range of abilities.

Using the link below will take you to our 2d shapes worksheet page where you will find a selection of our worksheets on 2d shapes for a variety of grades from kindergarten upwards.

If you are looking for 3d shape worksheets then you have found the right place.

All of our printable 3d shape worksheets from the website have been put onto the webpage below.

We have a wide selection of 3d shape sheets to cater for a range of grades and abilities.

There are sheets suitable from children age from kindergarten and up.

At Kindergarten level, the focus is on recognising 3d shapes and 2d shapes.

At 1st grade, we start identifying specific types of 3d shapes such as cones or prisms.

At 2nd grade we are beginning to name the shapes and count some of their faces.

At 3rd grade the focus turns to identifying properties such as faces, edges and vertices. We also start investigating the links between 3d shapes and their nets.

Symmetry Worksheets

Here is our selection of free printable symmetry worksheets for 2nd and 3rd grade.

The sheets are all graded in order from easiest to hardest.

• learn how to reflect simple shapes in a horizontal or vertical mirror line;
• learn how to reflect simple shapes in 2 mirror lines.

All the free printable geometry worksheets in this section support Elementary Math Benchmarks.

• Symmetry Worksheets - Block Symmetry

Geometry Nets Worksheets

On this page you will find information and worksheets about nets.

The worksheets consist of identifying and matching the net that matches the correct 3d shape.

• Geometry Nets Information and Worksheets

Geometry Riddles

Here you will find our free printable geometry riddles from 1st to 5th grade.

These riddles are all about problem solving with 2d shapes.

Using these riddles will help your child to:

• develop their geometry skills;
• develop their understanding of geometric language;
• apply their geometric knowledge to solve problems.

All the geometry riddles in this section support elementary math benchmarks.

• Geometry Worksheets (Riddles)

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GCSE Tutoring Programme

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In order to access this I need to be confident with:

This topic is relevant for:

Sophie Bessemer

Here we will learn about 2D shapes, including symmetry, perimeter, area, circles, sectors, arcs and angles in polygons.

There are also 2D shapes worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What are 2D shapes?

2D shapes are flat shapes which only have two dimensions; length and width.

Some examples of common 2D shapes names are triangles, rectangles, pentagons, hexagons, heptagons, octagons, nonagons, decagons and circles.

We can solve problems involving 2D shapes using a variety of methods.

Polygons are 2D shapes made from straight lines. You will deal with two different types of polygons.

Regular polygons

Regular polygons have specific properties. They have all sides of equal length and all interior angles are equal.

Examples of basic 2D shapes that are regular polygons are equilateral triangles and squares.

Irregular polygons

Irregular shapes (or polygons) do not have all equal sides and do not have all equal angles . When the number of sides is unknown, we describe this shape as an \textbf{n-gon} where the number of sides is given as \textbf{n}.

You will also need to be able to solve problems involving angles in polygons. The angles in a polygon can help determine whether a polygon is regular, a specific type of polygon or to determine how many sides a polygon has. You need to be familiar with interior and exterior angles of polygons.

Interior angles are angles that are contained within the polygon.

The sum of interior angles of any polygon can be calculated using the formula,

Sum of interior angles = (n-2) \times 180

where n represents the number of sides.

Exterior angles are supplementary to the interior angle. This means that the sum of the interior and exterior angles at a vertex always equals 180^{o}. We can use this property to find either the interior angle, or exterior angle at a vertex.

For any polygon the sum of exterior angles of a polygon 360^{o}.

Step-by-step guide: Polygons

Symmetry is when a line is drawn through a shape to make one side of the line a reflection of the other. It is a property of a 2D polygon or 3D polyhedron.

There are two different types of symmetry that you need to be aware of. Lines of symmetry and rotational symmetry .

For example, a rectangle has two lines of symmetry

and has order 2 rotational symmetry.

Step-by-step guide: Symmetry

Area is a measure of how much space there is inside of a 2 dimensional shape.

Here are formulae we need to remember to calculate the area of certain 2D shapes.

These are seen in the table below.

Step-by-step guide: How to work out area

The perimeter is the total distance around the outside of a 2D shape.

For example, let’s find the perimeter of the triangle.

For example, let’s find the perimeter of the rectangle.

The opposite sides of a rectangle are equal, a and b are the side lengths.

For example, let’s find the perimeter of the circle.

The perimeter of a circle is known as the circumference of a circle.

The circumference of this circle is

Step-by-step guide: How to work out perimeter

• Circles, sectors and arcs

Circles are round plane figures whose boundaries consist of points equidistant from a fixed point (the centre of the circle).

As well as the area and circumference of a circle we can also work out the following.

Step-by-step guide: Circles, sectors and arcs

• Parts of a circle

The parts of a circle have specific names and properties which you need to know for all circle related questions. An important fact to remember is that the radius of a circle is half of its diameter.

Step-by-step guide: Parts of a circle

• Area of a sector

The area of a sector is part of the area of a circle.

It can be found by using the formula \frac{\theta}{360} \times \pi r^{2}.

For example,

The area of this sector is

Step-by-step guide: Area of a sector

See also: Sector of a circle

The arc of a circle is part of the circle’s circumference.

It’s length can be found using the formula \frac{\theta}{360} \times \pi d.

The arc length of this sector is

Step-by-step guide: Arc length

See also: Arc of a circle

• Perimeter of a sector

To find the perimeter of a sector you need to find the arc length and then add it to the two straight sides which are both radii (i.e. the length around the outside of the sector).

The perimeter of this sector is

Step-by-step guide: Perimeter of a sector

• Segment of a circle

A segment of a circle is created by an arc length and a chord.

You may have to find its area using a combination of mathematical rules such as trigonometry or Pythagoras’ theorem.

Step-by-step guide: Segment of a circle

• Equation of a circle

The equation of a circle (at GCSE) can be given in the form below

Step-by-step guide: Equation of a circle

How to use 2D shapes

We can use 2D shapes in lots of different ways.

We will learn about:

Explain how to use 2D shapes

Area of 2D shapes worksheet

Get your free 2D shapes worksheet of 20+ area of 2D shapes questions and answers. Includes reasoning and applied questions.

2D shapes examples

Example 1: regular / irregular polygons.

The shape ABCDE below is made from two scalene triangles, and one isosceles triangle. M is a midpoint on the line CD . AM is a line of symmetry.

Determine what type of polygon ABCDE is and determine if it is regular or irregular.

• State / calculate the number of sides of the polygon.

The polygon ABCDE has 5 sides so it is a type of pentagon.

2 Determine the size of the angles / side lengths within the polygon.

As triangle ACD is isosceles and AM is a line of symmetry, angle ACD = angle ADC = 52^{o}.

As the sum of angles in a triangle is 180^{o},

angle CAD = 180-(52+52)=76^{o}.

As AM is a line of symmetry, triangles ABC and ACD must be congruent, sharing the same angles and side lengths. This means that,

• Angle BCA = angle ADE =56^{o}
• Angle DAE = angle CAB =16^{o}

Again, as the sum of angles in a triangle total 180^{o},

angle AED = 180-(56+16)=108^{o}.

This is the same for angle ABC as it is symmetrical to angle AED and therefore equal.

We now have all of the following angles,

3 Recognise the other properties of the polygon.

For a pentagon to be regular, all of the interior angles must be the same and side lengths must be the same.

Furthermore, as the interior angle sum of a pentagon is 540^{o}, each interior angle of a regular pentagon is equal to

Adding the angles at each vertex together, we have

Each interior angle of the polygon ABCDE is equal to 108^{o}.

However, the triangles AED and ABC are scalene. Therefore, AB has a different length to BC .

The interior angles are equal but the side lengths are not equal.

The polygon is an irregular pentagon.

Example 2: types of quadrilateral

Determine what type of quadrilateral ABCD is below.

Determine the size of the angles / side lengths within the quadrilateral.

All four side lengths are equal to 9cm with two pairs of parallel sides.

Angle BCD = 48+48=96^o and so ABCD cannot be a square.

CDE is a triangle. As the sum of angles in a triangle total 180^{o},

angle CED = 180-(42+48)=90^{o}.

Labelling this on the diagram, we have

Recognise the other properties of the polygon.

As vertically opposite angles are equal, angle AED = angle CED = 90^{o}.

As the sum of angles on a straight line total 180^{o},

Angle AED = 180-90=90^{o}.

This is also true for angle BEC .

We therefore have the diagonals intersecting at 90 degrees.

ABCD is a rhombus.

Remember: The properties of a rhombus are,

• Four equal sides
• Opposing sides are parallel
• Two opposing pairs of equal angles
• Diagonals bisect each other at 90 degrees
• Two lines of symmetry
• Rotational symmetry of order 2

Example 3: symmetry

State the number of lines of symmetry for a regular hexagon.

Locate the centre of the 2D shape.

To locate the centre of a shape with an even number of vertices, draw a pair of straight lines connecting two opposing vertices.

Use a ruler to visualise a horizontal and/or vertical line of symmetry through the centre of the shape.

The regular hexagon has a vertical line of symmetry.

The regular hexagon has a horizontal line of symmetry.

Continue to rotate the ruler around 180 degrees over the centre point to cover all sides and vertices.

Line of symmetry 3.

Line of symmetry 4.

Line of symmetry 5.

Line of symmetry 6.

A regular hexagon has 6 lines of symmetry.

Example 4: area of an isosceles triangle

Calculate the area of the triangle below.

Identify the base and perpendicular height of the triangle.

The two values that are perpendicular to one another are the 12cm along the base, and the vertical height of 8cm.

The 10cm is not required to find the area of this triangle.

Write the area formula.

The area of a triangle formula is,

\text{Area of a triangle}=\cfrac{\text{base }\times\text{height}}{2}.

Substitute known values into the area formula.

As the base = 12cm and the height = 8cm, we have

\text{Area of a triangle }=\cfrac{12\times{8}}{2}.

Solve the equation.

Write the answer, including the units.

As the units of length are in centimetres, the units of area are square centimetres.

The area of the triangle is 48cm^{2}.

Example 5: circumference of a circle

Calculate the circumference of a circle with a radius of 4cm. Write your answer correct to 2 decimal places.

Find the radius or diameter of the circle.

The radius of the circle is 4cm.

Use the relevant formula to calculate the circumference of the circle.

The formula for the circumference of a circle (C) in terms of the radius (r) is

Substituting r=4 into the formula, we have

\begin{aligned} C&=2\times\pi\times{4} \\\\ &=8\times\pi \\\\ &=8\pi \\\\ &=25.13274123 \\\\ &=25.13\text{ (2dp)} \end{aligned}

Give your answer clearly with the correct units.

The circumference of the circle is 25.13cm \ (2dp).

Example 6: perimeter of compound shapes – real life problem

A car park needs a new boundary fence installing. A sketch of the car park is given below.

Determine the length of the boundary wall (the car park entrance/exit must not be included in this value).

Add all the side lengths.

A few of the side lengths are missing and so we need to calculate these lengths first.

As AF = 8.8m and DE = 2.9m, the length of BC can be calculated by subtracting DE from AF ,

8.8-2.9 = 5.9m.

As AB = 12.4m and CD = 3.5m, the length of EF can be calculated by adding AB and CD together,

12.4+3.5=15.9m.

Writing these two measurements onto the diagram, we have

The perimeter is the sum of the side lengths and so we have the sum,

12.4+5.9+3.5+2.9+15.9+8.8=49.4m.

Note: For an L shape, this is the same as doubling the sum of the height and the width ((8.8+15.9)\times{2}=49.4m).

Write the final answer with the correct units.

As we need to calculate the boundary excluding the entrance/exit, we need to subtract the width of the entrance from the total perimeter.

49.4-2.4=47m

The perimeter of the car park excluding the entrance/exit is 47m.

Example 7: area of a sector

A circular cake is cut into 8 equal slices as shown below.

Calculate the area of the top of one slice of cake.

Find the length of the radius \textbf{r} .

The radius r = 9.5cm.

Find the size of the angle creating the sector

The angle of the sector is one eighth of a full turn. As a full turn is 360 degrees,

360\div{8}=45.

The angle of the sector is 45^{o}.

Substitute the value of the radius and the angle into the formula for the area of a sector.

The formula for the area of the sector is

\text{Area of a sector }=\frac{\theta}{360}\times\pi{r}^{2}

where \theta is the angle of the sector, and r is the radius of the overall circle.

Substituting \theta=45^{o} and r = 9.5cm, we have

\begin{aligned} \text{Area of a sector }&=\frac{45}{360}\times\pi\times{9.5}^{2} \\\\ &=\frac{1}{8}\times\pi\times{90.25} \\\\ &=11.28125\pi \\\\ &=35.44109212 \\\\ &=35.44\text{cm}^{2}\text{ (2dp)} \end{aligned}

Clearly state your answer.

The area of the top of the cake slice is 35.44cm^{2}.

Example 8: equation of a circle

Determine the equation of the unit circle, centred at the origin.

Write the general equation of a circle.

The general equation of a circle is x^{2}+y^{2}=r^{2}.

State any variables you know.

The radius of the unit circle is 1. We can also see this as the centre of the circle lies at the point (0,0) and a coordinate on the circumference of the circle is (1,0), giving us a radius of 1.

Substitute any values you know into the equation.

Substituting r=1 into the equation of a circle, we have

x^{2}+y^{2}=1^{2}.

Use the information you have to solve the problem.

Evaluating 1^{2} we get 1^{2}=1 \times 1=1.

Clearly state the answer.

The equation of the unit circle, centred at the origin is x^{2}+y^{2}=1.

Common misconceptions

• Rotational symmetry/lines of symmetry

Lines of symmetry are often confused with rotational symmetry. A line of symmetry on a two-dimensional shape divides the shape equally into two symmetrical pieces. Rotational symmetry is the number of times a shape fits into itself when rotated around its centre.

• Confusing perimeter with area

Remember, perimeter is distance around the outside of a shape, whilst area is the space inside the shape.

• Angles properties

Ensure you have a good understanding of the different types of angles (for example, acute angles and right angles) along with how to calculate angles in polygons and angles in parallel lines. Getting these confused can lead to misconceptions when problem solving.

Practice 2D shapes questions

1. Which of the following shapes has 2 lines of symmetry and rotational symmetry order 2.

Parallelogram

A parallelogram has no lines of symmetry and rotational symmetry order 2.

A square has 4 lines of symmetry and rotational symmetry order 4.

A kite has 1 line of symmetry and no rotational symmetry.

2. The rectangle has a perimeter of 42cm. Find x.

Form an equation for the perimeter.

Solve the equation, to find x=4.

3. Find the area of the isosceles triangle.

This is an isosceles triangle. We can split it vertically to form two identical triangles with base 6cm, and hypotenuse 10cm.

We then use Pythagoras’ theorem to find the height of the triangle.

Then we can use the formulae for the area of a triangle, and multiply the height by the base (remembering to use the base as 12cm for the area of the whole triangle) then divide by 2.

4. The sector has an area of \frac{27}{2} \pi cm^2, Find the perimeter in terms of \pi.

To find the perimeter of the sector we need to calculate the arc length. To do this we need to know the diameter of the sector.

We can work backwards using the formula for the area of a sector to find the radius to help us find the diameter.

Substituting in what we know,

Now we can calculate the arc length. If the radius is 9cm, the diameter must be 18cm.

For the total perimeter we need to add the two radii to the arc length.

5. What is the sum of the interior angles of a pentagon?

6. A regular polygon has an interior angle of 165^{\circ}. How many sides does it have?

The interior and exterior angle of a polygon sum to 180^{\circ}. We can find the exterior angle using this fact.

To find out the number of sides we then do,

2D shapes GCSE questions

1. The hexagon ABCDEF has one line of symmetry.

Angle FAB = angle ABC = 110^{\circ}.

Angle AFE = angle BCD .

Angle FED = angle CDE .

Angle CDE : angle BCD = 3 : 2.

Find the size of angle AFE .

Indicating sum of angles is 720^{\circ} or 540^{\circ} if line of symmetry used to form a pentagon.

Finding the sum of AFE and FED is 250^{\circ}.

Use of ratio 3:2 or sight of 3x and 2x.

Finding x=50.

Angle AFE = 100^{\circ}.

2. The perimeter of the rectangle is twice the perimeter of the isosceles triangle.

Find the area of the rectangle.

Correct expression for perimeter of rectangle or triangle,

Forming an equation linking the perimeters.

For example, 6x+2=2(2x+5) or equivalent.

Solving equation to get x=4.

Correctly substituted into length and width of rectangle.

Area given as 42cm^{2}.

3. The sector has a perimeter 25cm and radius 6cm.

Find the size of angle x. Give your answer to 3 significant figures.

Sight of arc length given as 13cm.

Process to use formula \frac{x}{360} \times 2 \times \pi \times 6=13

Learning checklist

You have now learned how to:

• Identify lines of symmetry and order of rotational symmetry in 2D shapes
• How to work out the perimeter
• Work out the area of triangles, quadrilaterals and circles
• Work out the area of compound shapes
• Identify and apply circle definitions and properties
• Calculate the circumference of a circle
• Calculate the area and arc length of a sector 
• Distinguish between regular and irregular polygon

The next lessons are

• Pythagoras theorem

Still stuck?

Prepare your KS4 students for maths GCSEs success with Third Space Learning. Weekly online one to one GCSE maths revision lessons delivered by expert maths tutors.

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Privacy Overview

A 2D shape is a flat shape that has only two dimensions - length and width, with no thickness or depth, that is the reason why it is called a two-dimensional shape . For example, a sheet of paper is two-dimensional in shape. It consists of a length and a width but does not have any depth or height. Some common 2D shapes are squares, rectangles, triangles, circles, and hexagons. Let us learn more about 2D geometric shapes , the difference between 2D and 3D shapes, along with some 2D shapes examples on this page.

What are 2D Shapes?

In geometry, 2D shapes (2 dimensional shapes) can be defined as plane figures that are completely flat and have only two dimensions - length and width. They do not have any thickness and can be measured only by the two dimensions. In comparison to these, a 3D (three-dimensional) shape has three dimensions - length, width, and height. For example, a dice is three-dimensional because it consists of a length, a width, and a height. Some common 3D shapes are cuboids, cones, pyramids, and cylinders.

2D Shapes Definition

A polygon is a 2 dimensional shape made up of straight line segments which are connected with each other, thus giving it a closed shape. A circle, square, rectangle, and triangle are some examples of two-dimensional shapes and these shapes can be drawn on paper. All the 2 dimensional shapes have sides, vertices (corners), and interior angles, except for the circle, which is a curved figure. 2D shapes with at least three straight sides are called polygons and these include triangles, squares, and quadrilaterals . The figure given below shows a list of 2D shapes that we commonly come across.

Types of 2D Shapes - Regular and Irregular 2D Shapes

A 2D shape can be classified as regular or irregular based on the length and the interior angles :

• A 2 dimensional shape (2D shape) is said to be regular if all its sides are equal in length and all its interior angles measure the same.
• A two dimensional shape (2D shape) is irregular if all the sides are of unequal length and all its angles are of unequal measures.

Observe the following figure which shows the difference between regular and irregular two dimensional shapes. It shows a list of 2d shapes.

Difference Between 2D and 3D Shapes

The following table shows a comparison between 2D and 3D shapes.

Properties of 2D Shapes

2-D shapes are flat and can be drawn on a sheet of paper. There are different types of regular and irregular two dimensional shapes like a circle, triangle, square, rectangle, pentagon, and hexagon. Let us learn about a few of them along with their properties.

2D Shapes Names

There are many 2d shapes in geometry. However, a few of them are commonly seen around us and are discussed below. Here is a list of 2D shapes examples along with their properties.

A circle is a closed 2 Dimensional shape made up of a curved line with no corners or edges. Some real-life examples of the circle are coins, wheels, and pizzas. A circle consists of various parts like the radius, diameter, circumference, and so on.

Properties of a Circle

Observe the properties of a circle to identify it as a two-dimensional shape.

• Circles are completely round and made up of a single curved line.
• The circumference is the length of the boundary of the circle.
• The radius is the distance from the center of the circle to the boundary of the circle.
• The diameter is a line segment that goes straight across the circle, through the center. It is the longest possible line that can be drawn inside a circle and is twice the length of the radius.

Observe the following figure to see the different parts of a circle.

A triangle is a 2 dimensional shape with three sides and three vertices (corners).

Properties of a Triangle

Observe the properties of a triangle to identify it as a two dimensional shape.

• A triangle is a closed shape with 3 sides, 3 vertices and 3 angles.
• It is a polygon whose interior angles add up to 180°.

Some real-life examples of a triangle are traffic signs, pyramids, and nachos. The following figure shows the sides and vertices of a triangle.

A square is a 2 dimensional shape with four equal sides and each angle is equal to 90˚. Some of the real-life examples of a square are: a loaf of bread and a chessboard.

Properties of a Square

Observe the properties of the given square PQSR to identify it as a 2D shape.

• All four sides are equal, i.e., side PQ = QS = RS = PR
• Side PQ is parallel to RS.
• Side PR is parallel to QS.
• All four internal angles measure 90 ̊.

A rectangle is a 2D shape with four sides in which the opposite sides are equal and parallel, and all the four angles measure 90 ̊. Some of the real-life examples of a rectangle are table tops, blackboards, cardboard, etc.

Properties of a Rectangle

Observe the properties of the given rectangle ABCD to identify it as a 2D shape.

• Side AB = DC
• Side AD = BC
• Side AB is parallel to DC.
• Side AD is parallel to BC.
• All four angles measure 90 ̊.

Area and Perimeter of 2D Shapes

The area of a 2D shape is the space enclosed within it. The perimeter of a 2D shape is the total length of its boundary. The following table shows the formulas that are used to calculate the area and perimeter of a few common 2D shapes:

Important Notes

Here are a few important notes that should be remembered while studying 2D shapes.

• Every 2D shape can be measured by its width and length.
• All the 2D shapes are completely flat.

2d Shapes Real Life Examples

A few real-life examples of 2d shapes are given below:

• A currency note which has the shape of a rectangle
• A chess board that has the shape of a square.
• A cookie in the shape of a circle.
• A honeycomb with little hexagons in it.

☛ Related Articles

• Geometric Shapes
• Plane Shapes
• Solid Shapes
• Visualizing Solid Shapes
• 2D and 3D Shapes Worksheets

2D Shapes Examples

Example 1: Does a rectangle come under the category of 2D shapes? The length of a rectangle is 32 m and the width is 20 m. Find the perimeter of this rectangle.

Solution : Yes, a rectangle comes under the category of 2D shapes. The length of the given rectangle = 32 m; width = 20 m

We know that the perimeter of a rectangle = 2 (length + width)

P = 2 (32 + 20) = 2 (52) = 2 × 52 = 104 m

Therefore, the perimeter of the rectangle is 104 m.

Example 2: Does a circle come under 2D shapes? Find the area of a circle whose radius is 6 cm.

Area of the circle = π × r 2

= 3.14 × 6 2

= 3.14 × 36

Therefore, the Area of the circle = 113.04 square cm

Example 3: Identify the 2D shapes from the following. Circle, rectangular box, Rubik's cube, hexagon

Among the given shapes, a circle and a hexagon are 2D shapes because they do not have any thickness or depth. A rectangular box and a Rubik's cube are 3D shapes because they have 3 dimensions, (length, width, and height).

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Practice Questions on 2D Shapes

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FAQs on 2D Shapes

What are 2d shapes in maths.

A 2D (two-dimensional) shape can be defined as a plane figure that can be drawn on a flat surface. It has only two dimensions - length and width, with no thickness or depth. Some of the basic 2D shapes are rectangle, pentagon, quadrilateral, circle, triangles , square, octagon , and hexagon.

What are 2D Shapes and 3D Shapes?

A flat shape that has two dimensions - length and width, is a 2D shape, whereas, a shape that has three dimensions - length, width, and depth (height) is a 3D shape .

What are the Properties of 2D Shapes?

A 2D shape is a flat shape that can be drawn on a plane surface. As the name suggests, it has only two dimensions of length and width without any thickness. While some 2D shapes have sides and vertices, others are made up of curved lines.

How many 2D Shapes are there?

There are many different kinds of 2D shapes, like the rectangle, pentagon, quadrilateral, circle, triangle, square , octagon and hexagon . These are flat shapes that can be drawn on a flat surface.

How to find the Area of 2D Shapes?

The area of a 2D shape is the space occupied by it. Since there are different kinds of 2D shapes like circles, squares, rectangles, etc., there are different formulas used to find their areas. For example, the area of a circle can be calculated with the help of the formula, Area = πr 2 , where 'r' is the radius of the circle and (pi) π is a constant with a value of 22/7. The area of a square is calculated with the formula, Area = side × side. The area of 2D shapes is expressed in square units.

How to Teach 2D Shapes?

Teaching 2D shapes is easy if the concept is introduced in the early years. First of all, the names of the basic shapes can be made to learn. Then, the students should be asked to identify the names of the shapes from a list of 2d shapes. Another interesting way can be by asking them to spot the shapes around them.

Name the 2D Shapes around the House.

Some commonly seen 2d shapes that can be seen in the house are, the rectangular surface of a notebook (rectangle), the circular shape of a clock (circle), the circular shape of a pizza, a square-shaped window (square)

Is Circle a 2D Shape?

Yes, a circle is a 2D shape because it exists on a plane with no depth. It is a curved shape that has no corners or edges.

Which 2D Shape has 4 Sides?

A quadrilateral is a 2D shape that has four sides. For example, quadrilaterals like squares and rectangles are 2D shapes with 4 sides.

Is Oval 2D or 3D Shape?

Oval is a 2D shape because it exists on a plane and has no depth. It is a curved figure with no edges or corners.

Problem solving with 2D shapes

I can use my understanding of 2D shapes to solve problems.

Lesson details

Key learning points.

• When problem solving, consider whether any elements are familiar from other areas of maths.
• Keep the goal in mind, it is easy to get distracted by too much information.
• With diagrams, it can be beneficial to add information to them (such as extra lines).

Common misconception

When finding the perimeter of a rectilinear compound shape, pupils might sum the perimeters of each rectangle.

Part of the perimeter for each rectangle is inside the compound shape. Therefore, not all of the perimeter of each rectangle contributes towards the perimeter of the compound shape in this problem.

Area - The area is the size of the surface and states the number of unit squares needed to completely cover that surface.

Compound shape - A compound shape is a shape created using two or more basic shapes.

Composite shape - A composite shape is an alternative for a compound shape.

This content is © Oak National Academy Limited ( 2024 ), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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2D Shapes Worksheets

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Exploring Geometry with 2D Shape Worksheets

Incorporating geometry worksheets with 2D figures into your math lessons is a fantastic way to teach young learners about basic shapes, their properties, symmetry, and more. These engaging, educational tools help children create a deep understanding of geometry, laying the groundwork for more advanced math concepts in the future.

Educational Benefits of 2D Shape Worksheets for Kindergarten

Geometry worksheets focusing on 2D figures are relevant and highly useful in educational settings for several compelling reasons.

• Foundation of Geometry: 2D figures serve as the foundational building blocks of geometry. Understanding these formations is essential for comprehending more advanced geometric concepts, such as angles, polygons, and spatial relationships.
• Visual Learning: Many students are visual learners, and 2D shape worksheets provide a visual means of teaching geometry. Visual aids, like diagrams, enhance comprehension and retention of geometric concepts.
• Hands-on Learning: A shape identification worksheet worksheet often includes hands-on activities that engage learners in drawing, coloring, cutting, and pasting a shape. These tactile experiences help solidify their understanding of a shape and its properties.
• Progressive Learning: Free printable 2D shapes worksheets for kindergarten can be tailored to different grade levels and abilities, from kindergarten to upper elementary grades. They can introduce a basic shape and gradually progress to more complex concepts, accommodating learners' developmental stages.
• Problem Solving: A worksheet can include problem-solving tasks that challenge learners to apply their knowledge of 2D figures in real-world scenarios. This encourages critical thinking and the practical application of geometry skills.
• Assessment and Feedback: Teachers can use two dimensional figures worksheets for formative and summative assessments to gauge students' understanding. A worksheet can provide opportunities to identify areas where your class may need additional support or enrichment.
• Differentiation: Worksheets on lines of symmetry in 2d shapes can be customized to meet the specific needs of diverse learners. Teachers can modify the complexity of questions or provide additional guidance to cater to individual learning styles and abilities.
• Home Learning: In addition to classroom use, 2 dimensional shapes worksheets are suitable for homework assignments and practice worksheets, allowing learning to reinforce shape recognition or identifying different shapes concepts, for example, and involve parents in the educational process.
• Creative Expression: A worksheet can incorporate creative activities, like shape art or shape stories, fostering artistic expression while reinforcing geometry concepts.
• Preparation for Advanced Math: Proficiency in 2D figures lays the groundwork for advanced math topics, such as geometry, trigonometry, and calculus. A strong foundation in geometry is essential for success in higher-level mathematics.

Worksheet Activity Ideas

• Shape Scavenger Hunt (Kindergarten): Have your class find everyday objects in their surroundings that match a given shape and create a personal shape collection.
• Shape Sorting (Shape Identification): Have your class cut out figures and sort them based on attributes like sides or angles, creating a labeled poster. Identifying shape worksheets are suitable for learners of various age groups and grade levels, making them adaptable to both early childhood education and elementary school curricula.
• Shape Art (Kindergarten): Young students draw creative pictures or designs using a basic 2D shape only, allowing them to practice fine motor skills through drawing.
• Shape Riddles (Kindergarten): Kids solve basic shape riddles on a worksheet by drawing or naming the matching shape.
• Shape Exploration: Kids arrange printable shapes to form a composite shape and describe attributes.
• Shape Story: Kids craft a short story where characters and objects are represented by a 2D shape.
• Shape Transformations: Have your class cut and rearrange shapes to create a new shape or patterns, discussing transformations.
• Symmetry Challenge (Lines of Symmetry in 2D Shapes): Kids identify and draw lines of symmetry on a shape, exploring the concept of symmetry.
• Shape Properties Poster (2D Shapes Properties Worksheets): Assign students a specific shape to make a poster showcasing its properties (e.g. how many sides). This is a great task that offers an opportunity to exercise creativity, while also helping a child learn about other shapes in a fun way through their peers' posters.
• Shape Attribute Search (2D Shapes Attributes Worksheets): Students find everyday objects or images that match given attributes (e.g. shapes with four equal sides) and paste them on a worksheet.

Steps to Make a 2D Shapes Worksheet

• Determine Learning Objectives: Clarify the specific two dimensional shapes or concepts you want to teach, such as identifying shapes, understanding properties, or exploring symmetry in a circle, square, oval, triangle, rectangles, or other quadrilaterals.
• Design Layout and Format: Create a clear and organized worksheet layout with a title, instructions, and space for student responses. Ensure that the format is age-appropriate and visually appealing.
• Select Shapes and Activities: Choose the 2D shapes or activities that align with your learning objectives. For example, decide if you want students to identify, draw, or calculate properties of shapes.
• Create Questions and Prompts: Write questions, prompts, or tasks that guide learners in working with the chosen shapes. Include a variety of activities to reinforce comprehension.
• Incorporate Visuals: Add images or illustrations of the 2D shapes you're focusing on. Visual aids help students better understand and recognize shapes.
• Proofread and Review: Carefully review the worksheet for accuracy, clarity, and age-appropriateness. Ensure that the questions and instructions are clear, and that the worksheet effectively reinforces the intended learning outcomes.

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• Geometric Solids Vocabulary Worksheets
• Area and Perimeter Worksheets
• 2D vs. 3D Worksheets
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Click "Save and Exit"

When you are finished, click this button in the lower right hand corner to exit your storyboard.

From here you can print, download as a PDF, attach it to an assignment and use it digitally, and more!

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Frequently Asked Questions about 2D Shapes Worksheets

How can i use 2d shapes worksheets to reinforce critical thinking skills in students.

To promote critical thinking, include problem-solving tasks in the worksheets. For example, ask students to calculate the perimeter or area of shapes, identify irregular shapes, or find patterns among shapes.

What are the key properties of 2D shapes, and how can I teach them?

Key properties of 2D shapes include the number of sides, types of angles, and symmetry. Teaching these properties can be done through worksheets that encourage students to count sides, identify angles, and explore lines of symmetry in shapes.

What are some creative ways to teach symmetry using worksheets?

You can teach symmetry using worksheets by providing shapes that students can complete to make them symmetrical. Additionally, you can offer worksheets with objects and ask students to draw the lines of symmetry themselves.

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Free printable 3rd Grade 2D shapes worksheets for math practice

Welcome to this amazing world of teaching 3 rd graders about 2D shapes. In this article, we will provide you with everything you need to know about 2D shapes for 3rd grade, including what they are, how to identify them, and how to compare and classify them. However, we encourage you to visit Mathskills4kids.com and find a variety of free printable 3rd Grade 2D shapes worksheets for extra math practice at home and in the classroom.

Are you looking for a fun and easy way to help your 3rd graders learn about 2D shapes ? Do you want to boost their math skills and confidence with engaging and interactive activities? If you answered yes to these questions, you are in the right place!

Download and use our worksheets right away!

Let’s get started!

Why 2D shapes matter for 3rd Graders

There are so many reasons why 2D shapes matter to 3 rd graders . First, they are everywhere in our world. They are the building blocks of many objects, patterns, designs, and art. Learning about 2D shapes helps 3rd graders develop spatial awareness, geometry, measurement, and problem-solving skills.

It also prepares them for more advanced math topics in the future, such as area, perimeter, fractions, and algebra.

An amazing unique thing about these Mathskills4kids’ 3 rd Grade 2D shapes worksheets is that they cover various 2D shapes concepts that, by the end of 3rd grade, students will be able to:

• Recognize and name common 2D shapes such as circles, squares, rectangles, triangles, pentagons, hexagons, octagons, etc.
• Describe the properties of 2D shapes such as sides, vertices (corners), angles, parallel and perpendicular lines, symmetry, etc.
• Compare and classify 2D shapes based on their properties and attributes .
• Identify and draw lines of symmetry in 2D shapes .
• Identify congruent (same size and shape) and similar (same shape but different size) 2D shapes .
• Identify and perform transformations such as reflection (flip), rotation (turn), and translation (slide) on 2D shapes.
• Apply 2D shapes knowledge to real-life situations and problems .

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What are 2d shapes, and how to identify them.

A 2D shape is flat with only two dimensions: length and width. It does not have any thickness or depth. You can draw a 2D shape on paper or a screen.

To identify a 2D shape , you need to look at its features, such as:

• The number of sides it has . A side is a straight or curved line that forms the edge of a shape.
• The number of vertices it has . A vertex (plural: vertices) is a point where two or more sides meet. It is also called a corner.
• The type and size of angles it has . An angle is the amount of turn between two sides. There are three types of angles : acute (less than 90 degrees), right (exactly 90 degrees), and obtuse (more than 90 degrees).
• The length of its sides . Some shapes have equal sides (all the same length), while others have unequal sides (different lengths).
• The relationship between its sides . Some shapes have parallel sides (always the same distance apart), while others have perpendicular sides (meet at right angles). Some shapes have both parallel and perpendicular sides.

Types of 2D Shapes and their Properties

There are many types of 2D shapes , but here are some of the most common ones that you will encounter in our free printable 3rd Grade 2D shapes worksheets math practice :

• Circle : A circle is round with no sides or vertices. It has one curved line that is always the same distance from the center. The distance from the center to any point on the circle is called the radius. The distance across the circle through the center is called the diameter. The distance around the circle is called the circumference.
• Square : A square is a four-sided shape with four equal sides and four right angles. It has four vertices. A square is a special type of rectangle that has all equal sides.
• Rectangle : A rectangle is a four-sided shape with two pairs of equal sides and four right angles. It has four vertices. A rectangle can be a square if all its sides are equal.
• Triangle : A triangle is a three-sided shape with three vertices and angles. The sum of the angles in any triangle is always 180 degrees. There are different types of triangles based on their sides and angles:
• Equilateral triangle : A triangle with three equal sides and three equal angles (60 degrees each).
• Isosceles triangle : A triangle with two equal sides and two equal angles opposite them.
• Scalene triangle : A triangle that has no equal sides or angles.
• Right triangle : A triangle with one right angle and two acute angles.
• Acute triangle : A triangle that has three acute angles.
• Obtuse triangle : A triangle with one obtuse angle and two acute angles.
• Pentagon : A pentagon is a five-sided shape with five vertices and five angles. The sum of the angles in any pentagon is always 540 degrees. There are different types of pentagons based on their sides and angles:
• Regular pentagon : A pentagon has five equal sides and five equal angles (108 degrees each).
• Irregular pentagon : A pentagon that has no equal sides or angles.
• Hexagon : A hexagon is a six-sided shape with six vertices and six angles. The sum of the angles in any hexagon is always 720 degrees. There are different types of hexagons based on their sides and angles:
• Regular hexagon : A hexagon has six equal sides and six equal angles (120 degrees each).
• Irregular hexagon : A hexagon that has no equal sides or angles.
• Octagon : An octagon is an eight-sided shape with eight vertices and eight angles. The sum of the angles in any octagon is always 1080 degrees. There are different types of octagons based on their sides and angles:
• Regular octagon : An octagon with eight equal sides and eight equal angles (135 degrees each).
• Irregular octagon : An octagon that has no equal sides or angles.

How to compare and classify 2D shapes

To compare 2D shapes , look at their properties and their similarities or differences. For example, you can compare two shapes by their number of sides, vertices, type, size of angles, length of sides, relationship between sides, etc.

To classify 2D shapes , you need to group them into categories based on their properties. For example, you can classify shapes by their number of sides (quadrilaterals, pentagons, hexagons, etc.), by their type of angles (right, acute, obtuse, etc.), by their symmetry (symmetrical, asymmetrical), etc.

Fun activities to practice 2D shapes skills

Learning about 2D shapes can be fun and engaging with these activities:

• Shape Hunt : Go on a shape hunt around your house or neighborhood and find as many 2D shapes as possible. Take pictures or draw them in a notebook. Label each shape with its name and properties.
• Shape Art : Use different 2D shapes to create your artwork. You can cut out shapes from paper or cardboard or use stickers, stamps, or magnets. Arrange the shapes into patterns, designs, or pictures. You can also use shapes to make letters, numbers, or words.
• Shape Puzzles : Cut out different 2D shapes from paper or cardboard and mix them up. Then try to put them back together to form larger shapes. To make shape puzzles, you can also use tangrams, pattern blocks, or geoboards.
• Shape Bingo : Make your bingo cards with different 2D shapes on them. Then call out the names of the shapes and mark them on your card. The first one to get a row, column, or diagonal wins!
• Shape Memory : Make pairs of cards with matching 2D shapes. Shuffle the cards and place them face down on a table. Take turns flipping over two cards at a time and try to find the matching pairs. If you find a pair, keep it and go again. If not, turn them back over and let the next player try. The one with the most pairs at the end wins!

Our selection of Mathskills4kids’ free printable 3 rd Grade 2D shapes worksheets for math practice

We have selected a variety of free printable 3 rd -grade 2D shapes worksheets from Mathskills4kids.com that will help your 3 rd graders practice their 2D shapes skills in a fun and engaging way. These worksheets cover different topics related to 2D shapes, such as:

• Identify two-dimensional shapes : In this worksheet, students will name the 2D shapes shown in the pictures. They can choose from a word bank that includes a circle, square, rectangle, triangle, pentagon, hexagon, octagon, trapezoid, parallelogram, and rhombus.
• Counting sides and vertices in 2-dimensional shapes : In this worksheet, students will count how many sides and vertices each 2D shape has and write the numbers in the boxes. They can use the shape names as a hint.
• Identify open and closed shapes : In this worksheet, students will identify which shapes are open and closed. They have to color the open shapes red and the closed shapes blue.
• Identifying a polygon : In this worksheet, students will identify which shapes are polygons and which are not. They have to color the polygons green and the non-polygons yellow.
• Identifying points, lines, line segments, and rays : In this worksheet, students will identify what kind of geometric element is shown in each picture. They have to write P for point, L for line, LS for line segment, or R for ray.
• Identifying angles greater than, less than, or equal to a right angle : In this worksheet, students will identify the angle type shown in each picture. They have to write > for greater than a right angle, < for less than a right angle, or = for equal to a right angle.
• Parallel, perpendicular, and intersecting lines : In this worksheet, students will identify the relationship between the pairs of lines shown in each picture. They will write P for parallel, PE for perpendicular, or I for intersecting.
• Reflection, rotation, and translation : In this worksheet, students will identify what transformation has been applied to each shape. They will write R for reflection, RO for rotation, or T for translation.
• How to identify congruent shapes : In this worksheet, students will identify which pairs of shapes are congruent and which are not. They have to color the congruent pairs green and the non-congruent pairs orange.
• Symmetry : In this worksheet, students will identify symmetrical shapes from asymmetrical ones. They have to draw a line of symmetry on each symmetrical shape and color it purple.

How to use 2D shapes worksheets effectively to boost 3rd graders’ math skills

These worksheets are designed to help third graders practice their 2D shapes skills in a fun and engaging way. However, they are not meant to replace direct instruction or hands-on activities. Here are some tips on how to use 2D shapes worksheets effectively to boost 3rd graders’ math skills :

• Before giving the worksheets to your students, review the concepts and vocabulary related to 2D shapes with them. You can use flashcards, posters, or interactive games to reinforce learning.
• Explain the instructions clearly and model how to complete each type of worksheet. You can use a projector or a smart board to show examples and demonstrate how to solve the problems.
• Provide feedback and guidance as your students work on the worksheets. You can check their answers with an answer key or use self-checking worksheets with QR codes or hidden messages.
• Encourage your students to use different strategies and tools to solve the problems. You can provide them with manipulatives such as geoboards or pattern blocks or let them use online tools such as virtual manipulatives or geometry apps.
• Differentiate the worksheets according to your student’s needs and abilities. You can modify the difficulty level, the number of problems, the type of questions, or the worksheet format to suit your student’s learning styles and preferences.
• Extend the learning beyond the worksheets by asking your students to apply their 2D shapes skills in real-life scenarios. You can ask them to find examples of 2D shapes in their environment, create them using different materials or design their worksheets for their peers.

Applying 2D shapes in real-life scenarios

One of the best ways to help your students understand and appreciate 2D shapes is to show them how they are used in real-life scenarios. Here are some examples of how you can connect 2D shapes to different subjects and topics:

• Art : You can ask your students to create artwork using different 2D shapes. You can inspire them with examples of famous artists who used geometric shapes in their paintings, such as Piet Mondrian, Wassily Kandinsky, or Paul Klee. You can also challenge them to create optical illusions or tessellations using 2D shapes.
• Science : You can ask your students to explore the properties and characteristics of 2D shapes with science concepts . For example, you can ask them to investigate how light reflects and refracts on different 2D shapes , how sound waves travel through different 2D shapes, or how forces affect the stability and strength of different 2D shapes .
• Social Studies : You can ask your students to identify and compare the 2D shapes used in different cultures and civilizations. For example, you can ask them to research how ancient Egyptians used 2D shapes in their architecture, art, and writing, how Native Americans used 2D shapes in their symbols and patterns, or how the Chinese used 2D shapes in their paper cutting and lantern making.
• Language Arts : You can ask your students to use 2D shapes as a creative writing prompt. For example, you can ask them to write a story based on a shape they draw or choose, a poem using shape words, or a riddle describing a shape without naming it.

Bonus: Find more resources to reinforce 2D shapes learning in Third Grade here!

If you are looking for more resources to reinforce 2D shapes learning in third grade , you are in luck! We have compiled a list of some of the best online resources that offer free games, videos, quizzes, and activities related to 2D shapes.

Here are some of our favorites:

• Math is Fun : This website has a lot of information and examples about 2D shapes, their properties, and how to classify them. You can also find interactive tools to explore shapes and measure angles. https://www.mathsisfun.com/geometry/plane-geometry.html
• Khan Academy : This online platform has a series of videos and exercises that cover 2D shapes and other geometry topics. You can watch the videos to learn new concepts, practice what you learned with quizzes, and track your progress. https://www.khanacademy.org/math/cc-third-grade-math/imp-geometry
• com : This website has a variety of games and worksheets that you can use to practice 2D shapes in a fun way. You can play matching games, puzzles, bingo, and more. You can also print out worksheets to color, cut, and glue shapes. https://www.education.com/resources/third-grade/geometry/

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Conclusion: Keep Practicing and Have Fun

We hope you enjoyed this article about free printable 3 rd -grade 2D shapes worksheets for math practice . We hope these worksheets will help your students improve their 2D shapes skills and have fun along the way.

Remember to check out our other articles and website, Mathskills4kids.com , for more free printable worksheets for third graders on different math topics. Happy learning!

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2d shape problem solving activity

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Age range: 7-11

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A great resource to reinforce the concepts linked to diagonals on 2-D shapes. Particularly liked the algebra extension activity. Will be using this resource with my Y5 class. Thank you.

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Manipulatives in Maths - A Classroom Guide for Teachers

Mathematical manipulatives are touted as essential tools for learning, but let's be honest—we've all experienced that moment of dread when we hand them out. Suddenly, your carefully planned lesson turns into chaos: One pupil starts building a fortress with the base ten blocks while another is hiding all the shiny counters.

Yet, despite these challenges, manipulatives play an important role in maths education. They bridge the gap between abstract concepts and tangible understanding, helping pupils grasp basic number sense. In fact, the National Curriculum emphasises their importance across all key stages, recognising that hands-on learning is vital for developing maths fluency, reasoning, and problem-solving skills.

So, how can we take advantage of these tools without losing control of the classroom? Let's explore the world of maths manipulatives—what they are, why they matter, and how to use them effectively in your primary school lessons.

What are manipulatives?

It can sound complicated, but manipulatives are simply hands-on tools that make abstract mathematical concepts concrete and visual . They're the building blocks, quite literally in some cases, that help pupils wrap their heads around tricky number ideas through good old-fashioned play, exploration, and modelling.

These learning aids come in all shapes and sizes, from the humble counter to the more elaborate Cuisenaire rods . Their key purpose? To give pupils something tangible to manipulate as they grapple with mathematical concepts. Whether it's using multilink cubes to understand place value or fraction circles to visualise parts of a whole, manipulatives help bridge the gap between 'maths on paper' and 'maths in real life'.

Common manipulatives you'll find in primary classrooms include:

Multilink cubes

Cuisenaire rods, base ten blocks, bead strings.

• Balance scales

Clock faces

Digit cards, hundred squares.

These tools align perfectly with the National Curriculum's aims of developing mathematical fluency, reasoning, and problem-solving skills. By allowing pupils to physically interact with mathematical ideas, manipulatives help build a strong foundation for more complex concepts down the line. They're not just toys or distractions—they're powerful learning tools that can transform how your pupils understand and engage with maths.

Why are they important?

Over the past two decades, research has consistently shown the positive impact of using manipulatives in the classroom. A 2013 report published in the Journal of Educational Psychology identified "statistically significant results" when teachers used manipulatives compared with when they only used abstract maths symbols. This highlights the role that manipulatives play in supporting conceptual understanding and facilitating the progression from concrete to abstract thinking.

Alignment with CPA approach

The NCETM agrees that physical manipulatives should play a central role in maths teaching. "Manipulatives are not just for young pupils, and also not just for those who can't understand something. They can always be of help to build or deepen understanding of a mathematical concept."

This approach aligns perfectly with the concrete-pictorial-abstract (CPA) progression. Once children are confident using manipulatives or 'concrete' resources, they can then move onto pictorial representations or the 'seeing' stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures , diagrams or models that represent the objects from the maths problem.

Enhance problem solving

But manipulatives do more than just support understanding—they're powerful tools for enhancing problem-solving skills. By allowing pupils to physically manipulate and visualise mathematical concepts, they can more easily devise strategies to tackle complex problems. This hands-on approach often leads to those 'aha!' moments we all love to see in our classrooms.

Support engagement

Moreover, manipulatives play an important role in fostering engagement and motivation. Let's face it—maths can sometimes seem dry and abstract to young learners. But introduce some colourful counters or interlocking cubes, and suddenly you've got a room full of eager mathematicians. This increased engagement is key to developing a positive attitude towards maths, which in turn supports long-term learning.

This deep understanding allows pupils to move beyond mere memorisation of facts and procedures, towards true mathematical fluency—where they can apply their knowledge flexibly and efficiently across a range of contexts.

In essence, manipulatives are not just helpful additions to our maths teaching toolkit—they're essential components in building a comprehensive, engaging, and effective mathematics education.

Types of manipulatives in primary mathematics

In this section, we'll break it common types of manipulatives into bite-sized pieces, just like we do for our pupils.

Physical manipulatives: the classics

These are the tangible, grab-them-with-your-hands resources that have been the backbone of maths classrooms for years. They're the ones that inevitably end up stuck between classroom seats and occasionally in someone's shoe.

Below is a list of common physical manipulatives in the classroom:

Ideal for teaching place value, addition, and subtraction with regrouping.

Fraction tiles

Excellent for comparing fractions and understanding equivalence.

Great for exploring 2D shapes, symmetry, and area.

Images: Wikipedia.org

Versatile tools for counting, measuring, and understanding volume.

Fantastic for developing number sense and exploring number relationships.

Essential for basic counting, sorting, and introducing simple addition and subtraction.

Useful for teaching multiplication, division, and fractions.

Image: Pinterest

Helpful for developing number sense and practicing skip counting.

Useful for probability exercises and generating random numbers for various activities.

Great for pattern recognition, matching, and basic addition facts.

Essential for teaching time-telling and understanding intervals.

Images: Pinterest & Pinterest

Useful for place value activities and forming large numbers.

Excellent for identifying number patterns and supporting multiplication and division.

Virtual manipulatives: a new kind of tool

Manipulatives have gone digital! These are interactive, online versions of our physical favourites. Think of them as the maths equivalent of e-books.

Some popular virtual manipulatives include:

Online number lines

These number lines are zoomable, clickable, and free of the uneven lines that are often result of our hand-drawn versions.

Digital base ten blocks

All the functionality without the risk of losing pieces under desks.

Interactive fraction tools

Slice and dice up pieces in any way imaginable.

Whether physical or virtual, the best manipulative is the one that helps your pupils understand the concept at hand. Whether that's a handful of multilink cubes or a fancy online simulator, if it's making those mathematical lightbulbs flicker on, you're on the right track!

Implementing manipulatives in the classroom - let them play!

Whether you have a bumper pack of manipulatives, a shared bank of resources or your very own DIY versions, it's important to teach children how to use them independently. Here are some best practices for integrating manipulatives effectively into your lessons:

• Introduce gradually : Bring in manipulatives one at a time. If you don't have enough for each child, set up a 'maths table' where pupils can take turns exploring. This works particularly well with younger years where 'choosing tables' are common.
• Allow for exploration : Give children a chance to play with and explore the manipulatives before using them for instruction. Through this exploration, they can start to imagine how the resource might be useful.
• How could you use this?
• How might this help you when adding or subtracting?
• Why do you think they're different sizes - what could that represent?
• Model usage : Once children are familiar with a resource, introduce a simple maths problem and ask them to use the manipulatives to solve it. Model the problem-solving process step-by-step, then guide children through it.
• Scaffold learning : Start with highly structured activities, then gradually reduce support as pupils gain confidence. For instance, begin with direct instruction on how to use base ten blocks for place value, then move to guided practice, and finally independent problem-solving.
• Year 1: Using counters or number lines to support addition and subtraction within 20.
• Year 2: Use fraction tiles to help pupils recognise, find, name and write fractions of a length, shape, set of objects or quantity.
• Year 3: Utilising place value charts (physical or digital) so pupils can recognise 3-digit numbers (100s, 10s and 1s).
• Integrate into lesson plans : Don't treat manipulatives as an add-on. Instead, weave them into your lessons as essential tools for understanding. Plan specific points in your lessons where manipulatives will be most beneficial.
• Support diverse learners : Manipulatives can be particularly helpful for English Language Learners (ELLs) and pupils with learning disabilities. They provide a universal language of mathematics that transcends verbal communication barriers.

Images: The Average Teacher

Manipulatives across Key Stages 1 and 2

Next, let's breakdown more examples of manipulatives in the classroom by Key Stage.

Key Stage 1 (Years 1-2): Laying the foundations

In these early years, it's all about getting hands-on with numbers and shapes.

• Number and Place Value : Introduce counters, number lines, and base ten blocks. Pupils can observe how 10 ones form a 'ten stick', helping them grasp place value concepts.
• Addition and Subtraction : Utilise multilink cubes for hands-on learning. Pupils can physically join or separate cubes to represent addition and subtraction operations.
• Fractions : Fraction tiles can be effective tools for teaching fractions. They provide a visual and tactile representation of concepts like 'half' and 'quarter'.
• Geometry : Employ geoboards for creating 2-D shapes. Pupils can then be asked to match these shapes on a 3-D surface to enhance spatial understanding.

Key Stage 2 (Years 3-6): Progressing with Purpose

As our mathematicians-in-training grow, so does the sophistication of our manipulatives. We're not ditching the basics, just building on them.

• Multiplication and Division : Array cards and Cuisenaire rods are useful for these operations. For multiplying by 6, pupils can line up 6 rods of 4 to visualise the concept.
• Fractions, Decimals, and Percentages : Fraction circles can be used alongside decimal place value charts. The 100 square is effective for teaching percentages.
• Geometry : The geoboard is a helpful tool for teaching perimeter, area, and symmetry concepts in a hands-on manner.
• Statistics : Data can be represented using multilink cube bar charts or human pictograms, making statistics more engaging for pupils.

CPA Journey: From Concrete to Pictorial to Abstract

Remember, our end goal is for pupils to solve problems without relying on physical props. Here's how we might progress:

• Concrete : Pupils physically manipulate objects to solve a problem. For example, using counters to work out 5 + 3.
• Pictorial : They draw a picture or diagram to represent the problem. Our 5 + 3 might become five circles and three circles.
• Abstract : Finally, they use mathematical symbols and numbers alone. "5 + 3 = 8."

The beauty of this approach? Pupils can always 'go back' a stage if they're struggling with a new concept. Stuck on an abstract problem? Draw a picture! Need more practise? Grab those counters!

Remember, every child's journey through these stages is unique. Some might race through, others might linger longer at certain points. The key is to ensure they have a solid understanding at each stage before moving on.

An example of moving from the concrete, to pictorial, to abstract stages.

Manipulative manners

Once you have introduced your resources, speak as a class and explain that they should come up with a set of rules for how they are treated and used. Giving children ownership over the manipulatives as well as the respect to make their own rules will make them feel accountable and lessen the likelihood of negative behaviours when using manipulatives. Write the rules up as a class and display them so they can be referred to.

Storing manipulatives

NRICH recommends children having access to manipulatives “Give open access to all the resources and allow the children free reign in choosing what to use to model any problem they may be tackling. I would make sure that children of all ages had this access from 3 to 11 years old and beyond.” While this is exactly what teachers would like to replicate in their classrooms, not all classes learn in the same way and this isn’t always achievable due to space, budgets and children’s prior experiences of manipulatives.

Once you have introduced a manipulative, decide as a class where you should store it . You know what works best for your class, so consider different options such as communal drawers, a maths table, individual packs or a collection of manipulatives for each table. Set clear rules around using and treating manipulatives to ensure they are not broken or lost. Additionally, you could create a monitor for each resource so the children can take ownership and make sure they stay tidy and accounted for.

Creating a classroom culture that uses manipulatives will aid children’s fluency and help develop their ability to solve problems, reason mathematically and share! If manipulatives are introduced in a considered and gradual way, with clear boundaries from an early age, children should see them as part of everyday learning and they will not be a novelty. They will be seen as tools instead of toys — and hopefully no more multilink towers!

For the community, by the community

Maths — No Problem! Community Event Conference, 28 November at The Royal Society, London.

Lily Lanigan

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A framework for solving parabolic partial differential equations

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Computer graphics and geometry processing research provide the tools needed to simulate physical phenomena like fire and flames, aiding the creation of visual effects in video games and movies as well as the fabrication of complex geometric shapes using tools like 3D printing.

Under the hood, mathematical problems called partial differential equations (PDEs) model these natural processes. Among the many PDEs used in physics and computer graphics, a class called second-order parabolic PDEs explain how phenomena can become smooth over time. The most famous example in this class is the heat equation, which predicts how heat diffuses along a surface or in a volume over time.

Researchers in geometry processing have designed numerous algorithms to solve these problems on curved surfaces, but their methods often apply only to linear problems or to a single PDE. A more general approach by researchers from MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) tackles a general class of these potentially nonlinear problems.  In a paper recently published in the Transactions on Graphics journal and presented at the SIGGRAPH conference, they describe an algorithm that solves different nonlinear parabolic PDEs on triangle meshes by splitting them into three simpler equations that can be solved with techniques graphics researchers already have in their software toolkit. This framework can help better analyze shapes and model complex dynamical processes.

“We provide a recipe: If you want to numerically solve a second-order parabolic PDE, you can follow a set of three steps,” says lead author Leticia Mattos Da Silva SM ’23, an MIT PhD student in electrical engineering and computer science (EECS) and CSAIL affiliate. “For each of the steps in this approach, you’re solving a simpler problem using simpler tools from geometry processing, but at the end, you get a solution to the more challenging second-order parabolic PDE.” To accomplish this, Da Silva and her coauthors used Strang splitting, a technique that allows geometry processing researchers to break the PDE down into problems they know how to solve efficiently.

First, their algorithm advances a solution forward in time by solving the heat equation (also called the “diffusion equation”), which models how heat from a source spreads over a shape. Picture using a blow torch to warm up a metal plate — this equation describes how heat from that spot would diffuse over it. 
This step can be completed easily with linear algebra.

Now, imagine that the parabolic PDE has additional nonlinear behaviors that are not described by the spread of heat. This is where the second step of the algorithm comes in: it accounts for the nonlinear piece by solving a Hamilton-Jacobi (HJ) equation, a first-order nonlinear PDE.  While generic HJ equations can be hard to solve, Mattos Da Silva and coauthors prove that their splitting method applied to many important PDEs yields an HJ equation that can be solved via convex optimization algorithms. Convex optimization is a standard tool for which researchers in geometry processing already have efficient and reliable software. In the final step, the algorithm advances a solution forward in time using the heat equation again to advance the more complex second-order parabolic PDE forward in time.


Among other applications, the framework could help simulate fire and flames more efficiently. “There’s a huge pipeline that creates a video with flames being simulated, but at the heart of it is a PDE solver,” says Mattos Da Silva. For these pipelines, an essential step is solving the G-equation, a nonlinear parabolic PDE that models the front propagation of the flame and can be solved using the researchers’ framework.

The team’s algorithm can also solve the diffusion equation in the logarithmic domain, where it becomes nonlinear. Senior author Justin Solomon, associate professor of EECS and leader of the CSAIL Geometric Data Processing Group, previously developed a state-of-the-art technique for optimal transport that requires taking the logarithm of the result of heat diffusion. Mattos Da Silva’s framework provided more reliable computations by doing diffusion directly in the logarithmic domain. This enabled a more stable way to, for example, find a geometric notion of average among distributions on surface meshes like a model of a koala. Even though their framework focuses on general, nonlinear problems, it can also be used to solve linear PDE. For instance, the method solves the Fokker-Planck equation, where heat diffuses in a linear way, but there are additional terms that drift in the same direction heat is spreading. In a straightforward application, the approach modeled how swirls would evolve over the surface of a triangulated sphere. The result resembles purple-and-brown latte art.

The researchers note that this project is a starting point for tackling the nonlinearity in other PDEs that appear in graphics and geometry processing head-on. For example, they focused on static surfaces but would like to apply their work to moving ones, too. Moreover, their framework solves problems involving a single parabolic PDE, but the team would also like to tackle problems involving coupled parabolic PDE. These types of problems arise in biology and chemistry, where the equation describing the evolution of each agent in a mixture, for example, is linked to the others’ equations.

Mattos Da Silva and Solomon wrote the paper with Oded Stein, assistant professor at the University of Southern California’s Viterbi School of Engineering. Their work was supported, in part, by an MIT Schwarzman College of Computing Fellowship funded by Google, a MathWorks Fellowship, the Swiss National Science Foundation, the U.S. Army Research Office, the U.S. Air Force Office of Scientific Research, the U.S. National Science Foundation, MIT-IBM Watson AI Lab, the Toyota-CSAIL Joint Research Center, Adobe Systems, and Google Research.

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• Leticia Mattos Da Silva
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• Department of Electrical Engineering and Computer Science

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I'm a parenting expert, and I want parents to let their children fail. Resilience is an important skill for all kids.

• As a parenting expert, I know instilling resilience in kids by letting them fail is important.
• To help them fail, parents should stop problem-solving for them.
• Parents should also ensure their kids don't have a negative mindset.

I'm a parenting expert, and I know raising children can be tricky, especially in today's accelerated, unpredictable, fear-based world. But challenges will be a regular part of their lives, and we must help kids become resilient so they can bounce back from those tough times and handle whatever comes their way.

In fact, I believe that kids whose parents do let them fail are more resilient, more confident , and ultimately more successful than kids with parents who swoop, hover, and rescue.

That's because each time we rescue, overprotect, or coddle, we send children a harmful message: "We don't believe you're capable of doing that alone." So, kids learn to depend on us to pick up their pieces, and there goes their chance to learn to bounce back .

The very next time you want to swoop in to help — don't. Instead, step back and subscribe to a new parenting behavior : "Never do for your child what your child can do for themself."

Here are five ways to do so from the book I wrote with Julia Cook titled "I Got This!"

Stop problem-solving

Jumping in to fix your kids' problems teaches them to expect you will always pick up the pieces and robs them of resilience.

Instead, teach your kids how to brainstorm so they can solve their own problems. Calmly say: "Name what's bothering you." Then, encourage your child to brainstorm solutions. You can even call it "The Solution Game."

With enough practice, kids can use brainstorming techniques to solve issues that creep up without our help.

Help them come up with a plan for the next time

If we want to raise world-ready, bounce-back kids, they must learn that setbacks are part of life.

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Instead, stay calm and help your child develop what I call a "Turn Around" plan. Ask: "What went wrong? What will you do differently next time?"

Set an example for them. When you mess up, show them what you learned and how you will change in the future. Resilient children recognize that mistakes are not life sentences, increasing their confidence.

Don't let them slip into negativity

Kids can be hard on themselves and can get stuck in negative thinking patterns that reduce their potential to bounce back.

Alternatively, help your child identify a positive statement to stay in control of tough situations.

Some ideas: "I'll get through it," "I got this," or "It's OK; I'll get it."

Encourage kids to repeat their chosen upbeat phrase until they can use it on their own.

Tip: You might repeatedly use the phrase until your outer voice becomes your child's inner voice that helps them rebound.

Break down goals, so they're more manageable

Some kids give up because they feel overwhelmed. Reducing tasks into smaller, manageable parts also reduces worries and boosts children's confidence .

Suppose your child is overwhelmed with their math homework. Instead of coddling them, tell your kid to "chunk your worry" by "covering all your math problems with paper except the top row. Then, as each row is completed, lower the paper down to the next row and the next."

Suppose your kid fears swimming. You can say: "Today, put your big toe in the water." The next day, they can put in an ankle, then a knee, and finally their whole body. With each phase, they can feel a sense of accomplishment.

Teach your kid to regulate their response

Taking slow, deep breaths and then exhaling twice as long as the inhale gets oxygen to the brain and can help kids stay in control-especially during hard times.

Instead of reacting to your kid's stress, say: "Just breathe! Take a big inhale from down in your tummy and exhale from your lips."

Teach younger children with soap bubbles. Tell them: "See how slowly and far you can blow your worries away without popping the bubble."

With enough practice, kids will learn to become their own regulators and be calmer when challenges arise.

The good news is that resilience is not locked into DNA, ZIP codes, or temperament. And it's never too late or early to help children learn to rebound. Doing so may be their best hope to handle an unpredictable world.

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