# 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

## 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.

## 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.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

Step-by-step guide: Area of a triangle

### Example 3: area of a parallelogram

Find the area of the parallelogram below.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

Step-by-step guide: Area of a parallelogram

### Example 4: area of a trapezium

Find the area of the trapezium below.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

Step-by-step guide: Area of a trapezium

### Example 5: area of a circle

A circle has a diameter of 8mm.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

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.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

### 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.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

### 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.

Write down the formula for the area of the shape.

Identify the values required for the formula.

Substitute the values and calculate.

### 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

7.5cm^2

6cm^2
\begin{aligned} A&=\frac{1}{2} \times \text{base} \times {height} \\\\ &=\frac{1}{2}\times 4\times 3\\\\ &=6 \mathrm{~cm}^{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}^{2} \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}^{2} \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

## Still stuck?

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