# Area

Here we will learn about area, including how to calculate the area of triangles, quadrilaterals, circles and compound shapes.

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

## What is area?

Area is a measure of the amount of space there is inside a 2D shape. The area of a shape is measured in square units such as cm^{2}, mm^{2} or m^{2}.

To calculate area we can use different formulas depending on the shape.

Below are the formulas to find the area of some common shapes. For each of the shapes below there are also pages where you can see more examples.

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

We can also find the area of polygons and compound shapes using a combination of these formulas.

A compound shape can be split into two or more smaller shapes and the area of these shapes can be found. The total area of the compound shape can be found by adding the smaller areas together.

Here are some examples of compound shapes.

Step-by-step guide: Area of compound shapes

### What is area? ## How to calculate area

In order to calculate the area of a shape:

1. Write down the formula for the area of the shape.
2. Identify the values required for the formula.
3. Substitute the values and calculate.

### Explain how to calculate area ## Area examples

### Example 1: area of a rectangle

Find the area of this rectangle below.

1. Write down the formula for the area of the shape.

Rectangle,

\text { Area }=\text { base } \times \text { height } .

2Identify the values required for the formula.

Base = 7m

Height = 4m

3Substitute the values and calculate.

\begin{aligned} \text { Area } &=\text { base } \times \text { height } \\\\ \text { Area } &=7 \times 4 \\\\ &=28 \end{aligned}

In this case we are working with metres so our final answer must be in square metres.

\text { Area }=28 \mathrm{~m}^{2}

Step-by-step guide: Area of a rectangle

### Example 2: area of a triangle

Find the area of the triangle below.

Triangle,

A=\frac{1}{2} \times \text{base} \times \text{height} .

Base = 8cm

Height = 6cm

\begin{aligned} A&=\frac{1}{2} \times \text{base} \times \text{height}\\\\ &=\frac{1}{2} \times 8 \times 6\\\\ &=24 \end{aligned}

In this case we are working with cm so our final answer must be in square cm.

\text { Area }=24 \mathrm{~cm}^{2}

Step-by-step guide: Area of a triangle

### Example 3: area of a parallelogram

Find the area of the parallelogram below.

Parallelogram,

\text { Area }=\text { base } \times \text { height } .

Base = 6cm

Perpendicular height = 30mm

We need all units to be the same to calculate the area. Here it is easiest to change 30mm to 3cm.

Height =3cm

\begin{aligned} \text { Area }&= \text { base } \times \text { height } \\\\ &=6 \times 3 \\\\ &=18 \end{aligned}

\text { Area }=18 \mathrm{~cm}^{2}

Step-by-step guide: Area of a parallelogram

### Example 4: area of a trapezium

Find the area of the trapezium below.

Trapezium,

\text { Area }=\frac{1}{2}(a+b)h .

a and b are the parallel sides and h is the height of the trapezium.

a=5 and b=9

h=4

\begin{aligned} \text{Area }&=\frac{1}{2}(a+b)h\\\\ &=\frac{1}{2}\times(5+9)\times 4\\\\ &=\frac{1}{2} \times 14 \times 4\\\\ &=28 \end{aligned}

\text {Area }=28 \mathrm{~m}^{2}

Step-by-step guide: Area of a trapezium

### Example 5: area of a circle

A circle has a diameter of 8mm.

Circle,

\text{Area} = \pi r^{2} .

r is the radius of the circle. In this question we are given the diameter, 8mm. The radius is half of the diameter which is 4mm.

\begin{aligned} \text{Area} &= \pi r^{2}\\\\ &= \pi \times 4^{2}\\\\ &=50.26548246 \end{aligned}

The question asks for the answer to be given to one decimal place so we need to round the answer.

\text { Area }=50.3 \mathrm{~mm}^{2}

Step-by-step guide: Area of a circle

### Example 6: area of a compound shape

This shape is made from a square and a semicircle.

Find the area of the shape. Give your answer to 2 decimal places.

Break down the compound shape into basic shapes.

The shape can be split into a square with side length 6cm and a semicircle with radius 3cm.

Square,

\text{Area of square} = \text{base} \times \text{height} .

Semi-circle,

\text{Area of a circle}=\pi r^{2} .

\text{Area of semi-circle }=\text{Area of circle } \div 2

Square,

Base =6cm, height =6cm .

Semi-circle,

Square,

\text{Area of square} = \text{base} \times \text{height}

Base =6cm, height =6cm

\begin{aligned} \text{Area of square}&=6 \times 6\\\\ &=36 \mathrm{~cm}^{2} \end{aligned}

Semi-circle,

\text{Area of a circle}=\pi r^{2}

\begin{aligned} \text{Area of a circle }&=\pi \times 3^{2}\\\\ &=28.27433388 \end{aligned}

\begin{aligned} \text{Area of semi-circle }&=\text{Area of circle } \div 2\\\\ &=28.27433388 \div 2\\\\ &=14.13716694 \end{aligned}

Total Area =36+14.13716694=50.13716694

Total area = 50.14cm^2

### Example 7: using the area to find a length

The area of this triangle is 30cm^{2}. Work out the length of the side labelled x.

Triangle,

A=\frac{1}{2} \times \text{base} \times \text{height} .

Area = 30cm^2

Height = 6cm

\begin{aligned} A&=\frac{1}{2} \times \text{base} \times \text{height}\\\\ 30&=\frac{1}{2} \times x \times 6\\\\ 30&=3x\\\\ 10&=x \end{aligned}

In this case we are working with cm and square cm so our final answer must be in cm.

x=10cm

### Example 8: using the area to find the radius

The area of this circle is 25 \pi m^{2}.

Work out the radius of the circle.

Circle,

A=\pi r^{2} .

\text{Area }=25 \pi \mathrm{~m}^{2}

\begin{aligned} A&=\pi r^{2}\\\\ 25 \pi &=\pi r^{2}\\\\ 25&=r^{2}\\\\ \sqrt{25}&=r\\\\ 5&=r \end{aligned}

### Common misconceptions

• Using an incorrect formula

There are several different formulae for the different shapes – make sure you use the correct one.

• Using the incorrect unit of area or not including units

Area calculations are measured in square units.

For example, square millimetres, square centimetres, square metres, square kilometres, square inches, square feet, square yards, square miles etc.

• Confusing perimeter, area and volume

Two dimensional shapes have a perimeter and an area. Three dimensional figures have a surface area and a volume. You need to make sure you are calculating the correct measurement required for each question.

• Calculating with different units

All measurements must be in the same units before calculating an area or surface area.

For example, you can’t have some measurements in cm and some in m.

To work out the area of a circle we need the radius, which is the distance from the centre of the circle to the edge of the circle.

### Practice area questions

1. Find the area of the rectangle. 11m^2 22m^2 24m^2 12m^2 Use the formula

\text{Area }= \text{ base } \times \text{ height}

8 \times 3=24\mathrm{~cm}^{2}

2. Find the area of the triangle. 12cm^2 10cm^2 7.5m^2 6m^2 \begin{aligned} A&=\frac{1}{2} \times \text{base} \times {height} \\\\ &=\frac{1}{2}\times 4\times 3\\\\ &=6 \mathrm{~m}^{2} \end{aligned}

3. Find the area of the parallelogram in m^{2}. 36m^2 41.76m^2 29m^2 3600m^2 Convert 720cm into metres by dividing by 100.

\text{Base }=720 \div 100=7.2

\text{Area }= \text{ base } \times \text{ height}

Careful not to multiply by the length of the diagonal instead of the perpendicular height.

\begin{aligned} \text{Area }&=7.2 \times 5\\\\ &=36\mathrm{~m}^{3} \end{aligned}

4. Find the area of the trapezium below in cm^{2}. 6020cm^2 5250cm^2 4480cm^2 5762cm^2 Convert 0.64m into 64cm and 860mm into 86cm.

\begin{aligned} \text{Area }&=\frac{1}{2}(a+b)h\\\\ &=\frac{1}{2} \times (64+86) \times 70\\\\ &=\frac{1}{2} \times 150 \times 70\\\\ &=5250 \mathrm{~cm}^{3} \end{aligned}

5. A circle has an area of 64 \pi \mathrm{~cm}^{2}.

What is the diameter of the circle?

4cm 8cm 16cm 32cm \begin{aligned} \text{Area }&=\pi r^{2}\\\\ 64 \pi &= \pi r^{2}\\\\ 64&=r^{2}\\\\ \sqrt{64}&=r\\\\ 8&=r \end{aligned}

6. Find the area of the shape. 88cm^2 66cm^2 110cm^2 93.5cm^2  \begin{aligned} \text{Area of rectangle }&=\text{ base } \times \text{ height}\\\\ &=11 \times 6\\\\ &=66 \mathrm{~cm}^{2} \end{aligned}

\begin{aligned} \text{Area of triangle }&=\frac{1}{2} \times \text{ base } \times \text{ height }\\\\ &=\frac{1}{2} \times 11 \times 4\\\\ &=22 \mathrm{~cm}^{2} \end{aligned}

Total area: 66+22=88cm^2

### Area GCSE questions

1. A square has a perimeter of 28cm. Find its area.

(2 marks)

Side length = 28\div 4 =7 \mathrm {~cm}

(1)

49cm^2

(1)

2. These two shapes have the same area. Work out the height of the rectangle. (3 marks)

Area of trapezium = \frac{1}{2} \times (6+10) \times 5 = 40 \mathrm{~cm}^{2}

(1)

Area of rectangle = 40cm^2

(1)

Height of rectangle = 40 \div 4 = 10 \mathrm{~cm}

(1)

3. Archie needs to paint a wall with two coats of paint.

The wall is pictured below with its measurements in metres. A tin of paint covers 20m^{2}. How many tins of paint does Archie need for the two coats of paint?

(4 marks)

Splitting the shape into two or more shapes, e.g. a rectangle and triangle or a trapezium and rectangle.

(1)

Finding the area of one of the shape

(1)

Correct total area of 38.5m^2

(1)

Correct number of tins = 4

(1)

## Learning checklist

You have now learned how to:

• Work out the area of triangles, quadrilaterals and circles
• Work out the area of compound shapes

## The next lessons are

• How to work out perimeter
• Symmetry
• Circles, sectors and arcs

## Still stuck?

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#### FREE GCSE Maths Practice Papers - 2022 Topics

Practice paper packs based on the advanced information for the Summer 2022 exam series from Edexcel, AQA and OCR.

Designed to help your GCSE students revise some of the topics that will come up in the Summer exams.