Area of a triangle

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GCSE Maths Geometry and Measure Area

Area Of A Triangle

Here we will learn about the area of a triangle, including how to find the area of a triangle with given dimensions.

There are also area of a triangle 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 the area of a triangle?

The area of a triangle is the amount of enclosed two-dimensional space that three connected straight lines take up.

In order to derive the area of a triangle, we start with the area of a rectangle.

The area of the rectangle below would be calculated by multiplying the base by the height ( $b \times h$ ). This is usually pronounced as “base times height”.

We can split a rectangle into two smaller rectangles by drawing a perpendicular line through any two parallel sides of the rectangle.

Splitting these two rectangles in half by joining the diagonals together, we can create two pairs of congruent (identical) right-angled triangles.

The area of the rectangle is now halved to get the area of the triangle, so the area of the triangle is exactly half the area of the rectangle, when we know the base and the perpendicular height of the triangle.

What is the area of a triangle?

Area of a right angled triangle formula

The formula for the area of a triangle is:

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

This can be shortened to

A=\frac{1}{2}bh

where $b$ is the base length and $h$ is the perpendicular height of the triangle.

(Perpendicular means that the base and the height are at right angles to each other.)

Your final answer must be given in square units, e.g. $cm^2, \ m^2,$ or $mm^2$ .

Higher – There is an alternative method to find the area of a triangle which involves sine, two sides and the included angle. This is part of trigonometry.

\text{Area of a triangle}=\frac{1}{2}ab\sin{C}

Step-by-step guide: Area of a triangle ( $\frac{1}{2}ab \sin C$ )

On this page we’ll look at the area of scalene triangles. A scalene triangle is a triangle where all of the sides and all of the angles are different.

This means that a scalene triangle can contain three acute angles, a right-angle, or one obtuse angle (and two acute angles).

Remember: All of the angles within any triangle must add up to $180^{\circ}$ .

Labelling triangles

The triangle below has the three sides labelled $a, b,$ and $c,$ and the three vertices labelled $A, B,$ and $C.$ The side $a$ opposes the vertex $A$ and so on.

Area of a triangle Featured Card Image 1

You should also be familiar with the notation that this triangle is called triangle $ABC$ .

Each side could also be labelled using the capital letters that denote the start and end point of the line. For example, the side $c$ is also known as the side $AB$ , the side $a$ can be referred to as side $BC$ , and side $b$ can be expressed as side $AC$ .

How to calculate the area of a triangle

In order to calculate the area of a triangle:

Identify the base and perpendicular height of the triangle.
Write the area formula.
Substitute known values into the area formula.
Solve the equation.
Write the answer, including the units.

How to calculate the area of a triangle

Area of a triangle worksheet

Get your free area of a triangle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Area of a triangle worksheet

Get your free area of a triangle worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Related lessons on area

Area of a triangle is part of our series of lessons to support revision on area. You may find it helpful to start with the main area lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Area of a triangle examples

Example 1: right-angled scalene triangle

Calculate the area of the right-angled triangle $ABC$ . Write your answer in square centimetres.

Identify the base and perpendicular height of the triangle.

The sides AC and BC are perpendicular to each other (the angle at $C$ is a right-angle) and so we can state the values for the base and the perpendicular height of the triangle to be:

The base: $b=3.2 \ cm$

The perpendicular height: $h=5.3 \ cm$

2Write the area formula.

A=\frac{1}{2}bh

3Substitute known values into the area formula.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(3.2)(5.3) \end{aligned}

4Solve the equation.

Work out the value for $A$ by solving the equation:

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(3.2)(5.3) \\\\ &=\frac{1}{2}\times 3.2\times 5.3\\\\ &=8.48 \end{aligned}

5Write the answer, including the units.

A=8.48\text{ cm}^2

Remember: Your final answer must be in square units.

Example 2: obtuse scalene triangle

For triangle $ABC$ , the angle at $C$ is an obtuse angle, the length $AC=12cm$ , and the height of the triangle is $9cm$ . Calculate the area of triangle $ABC.$

Identify the base and perpendicular height of the triangle.

As AC is perpendicular to the height stated for the triangle, we can state:

$\begin{aligned} &b=12 \ cm \\\\ &h=9 \ cm. \end{aligned}$

Write the area formula.

A=\frac{1}{2}bh

Substitute known values into the area formula.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(12)(9) \end{aligned}

Solve the equation.

Solving for $A$ , we have

$\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(12)(9) \\\\ &=\frac{1}{2}\times 12\times 9\\\\ &=54 \end{aligned}$

Write the answer, including the units.

A=54\text{ cm}^2

Example 3: acute scalene triangle – mixed units

Calculate the area of triangle $ABC$ below. Write your answer in square metres.

Identify the base and perpendicular height of the triangle.

Currently we have $2$ different units here. We must convert them to a common unit (here we will use metres as the question requires the answer to be in square metres):

$240 \ cm = 2.4 \ m.$

So the base and the perpendicular height of the triangle are:

$\begin{aligned} &b=2.4 \ m \\\\ &h= 2\ m \end{aligned}$

Write the area formula.

A=\frac{1}{2}bh

Substitute known values into the area formula.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(2)(2.4) \end{aligned}

Solve the equation.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(2)(2.4) \\\\ &=\frac{1}{2}\times 2\times 2.4\\\\ &=2.4 \end{aligned}

Write the answer, including the units.

A=2.4\text{ m}^2

Example 4: obtuse scalene triangle – too much information

$PQR$ is an obtuse scalene triangle. The point $S$ lies on the line $PQ$ such that angle $PSR=90^{\circ}$ .

Calculate the area of the triangle $PQR$ below, correct to $2$ decimal places.

Identify the base and perpendicular height of the triangle.

Here, we need to distinguish which two lengths are perpendicular to one another as we can use these as the base and the height of the triangle $PQR$ .

As the question states that angle $PSR=90^{\circ}$ , the side of the triangle $PQ$ is perpendicular to the length $RS$ which is the height of the triangle. We therefore can state:

$\begin{aligned} &b= 11.02 \ cm \\\\ &h= 4.48\ cm. \end{aligned}$

Write the area formula.

A=\frac{1}{2}bh

Substitute known values into the area formula.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(11.02)(4.48) \end{aligned}

Solve the equation.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(11.02)(4.48)\\\\ &=24.6848 \end{aligned}

Write the answer, including the units.

A=24.68 \text{ cm}^2 \text{ (2dp)}

Example 5: compound shape

A rectangle and a triangle are placed together to make a compound shape $ABCDE$ .

Calculate the area of the compound shape to $1$ decimal places.

Identify the base and perpendicular height of the triangle.

If we split the compound shape into two polygons, we have a rectangle and a triangle.

As $AB$ and $DE$ are two parallel sides of a rectangle, $AB=DE$ and so we can calculate the vertical height of the triangle by subtracting $4m$ from $6m.$ This gives us the following dimensions on the original diagram:

Area of a triangle Example 5 Step 1 Image 1

If we sketch another copy of the triangle and label the dimensions of it, we get:

Area of a triangle Example 5 Step 1 Image 2

We can now state the values for the base and perpendicular height:

$\begin{aligned} &b= 5 \ m \\\\ &h= 2 \ m. \end{aligned}$

Write the area formula.

A=\frac{1}{2}bh

Substitute known values into the area formula.

\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(5)(2) \end{aligned}

Solve the equation.

Work out the calculation

$\begin{aligned} A &=\frac{1}{2}bh \\\\ &= \frac{1}{2}(5)(2)\\\\ &=\frac{1}{2}\times 5\times 2\\\\ &=5 \end{aligned}$

Now we must find the area of the rectangle.

$\begin{aligned} A &=l\times w \\\\ &= 5\times 4\\\\ &=20 \end{aligned}$

$\text{Total area}=5+20=25$

Write the answer, including the units.

A=25\text{ m}^2.

How to find a missing side length given the area

Sometimes a question might give you the area and ask you to work out the missing length. In order to do this you must rearrange the formula to make the missing value the subject, and then solve for this letter.

In order to find a missing length given the area:

Write the area formula.
Substitute the known values into the area formula.
Rearrange the equation and solve.
Write the answer, including the units.

Missing side length example

Example 6: calculating the base length given the area

The area of a triangle is $45 \ cm^2$ . The height of the triangle is $10 \ cm.$ Find the length of the base of the triangle.

Write the area formula

A=\frac{1}{2}bh

Substitute the known values into the area formula

\begin{aligned} A&=\frac{1}{2}bh\\\\ 45&=\frac{1}{2}\times b\times 10 \end{aligned}

Rearrange the equation and solve.

We can solve the equation by rearranging to find the unknown $b.$

First we can tidy up the right-hand-side

$\begin{aligned} 45&=\frac{1}{2}\times b\times 10\\\\ 45&=b\times 5 \end{aligned}$

Then we divide both sides by $5.$

$\begin{aligned} 45&=b\times 5\\\\ \frac{45}{5}&=b\\\\ b&=9 \end{aligned}$

Write the answer, including the units.

b=9\text{ cm}

Remember: Your final answer must be in linear units only as it is a length.

Common misconceptions

Identifying the correct information to use

The base and the height must be at a right angle to each other. They must be perpendicular. You may be given additional sides of a triangle that are not needed. You may also need to rotate the triangle around so that you can identify the base and height more easily.

Area of a triangle Common Misconceptions Image

Units

It is common to forget the units for area in the final answer. When calculating area, your answer must always have units squared (or square units).

Practice area of a triangle questions

8\text{ cm}^2

7.5\text{ cm}^2

15\text{ cm}^2

16\text{ cm}^2

The base and the perpendicular height are $5 \ cm$ and $3 \ cm$ respectively, so the calculation we need to do is:

\begin{aligned} A&=\frac{1}{2}bh \\\\ &=\frac{1}{2}(5)(3) \\\\ &=\frac{1}{2}\times 5\times 3 \\\\ &=7.5 \end{aligned}

The units will be $cm^2.$

126\text{ cm}^2

252\text{ cm}^2

12.6\text{ cm}^2

25.2\text{ cm}^2

The base is $63 \ mm$ which needs to be converted to $6.3 \ cm.$ The height is $4 \ cm.$ The calculation we need to do is:

\begin{aligned} A&=\frac{1}{2}bh \\\\ &=\frac{1}{2}(6.3)(4) \\\\ &=\frac{1}{2}\times 6.3\times 4 \\\\ &=12.6 \end{aligned}

The units will be $cm^2.$

24\text{ cm}^2

21\text{ cm}^2

12\text{ cm}^2

10.5\text{ cm}^2

It may be easier to see the base and height if you turn the triangle. The base and the height are $3 \ cm$ and $7 \ cm.$ The $8 \ cm$ is not needed. So the calculation we need to do is:

\begin{aligned} A&=\frac{1}{2}bh \\\\ &=\frac{1}{2}(3)(7) \\\\ &=\frac{1}{2}\times 3\times 7 \\\\ &=10.5 \end{aligned}

The units will be $cm^2.$

13.8 \text{ m}^2

13 \ 800 \text{ cm}^2

6.9\text{ m}^2

69 \ 000\text{ cm}^2

The base is $460 \ cm$ which needs to be converted to $4.6 \ m.$ The height is $3 \ m.$ The calculation we need to do is:

\begin{aligned} A&=\frac{1}{2}bh \\\\ &=\frac{1}{2}(4.6)(3) \\\\ &=\frac{1}{2}\times 4.6 \times 3 \\\\ &=6.9 \end{aligned}

The units will be $m^2.$

1250\text{ m}^2

500\text{ m}^2

750\text{ m}^2

2000\text{ m}^2

The compound shape is made up of a rectangle and a triangle.

Area of a triangle Practice Question 5 Explanation Image

The area of the rectangle is $10\times{50}=500\text{ m}^2$

The area of the triangle is:

\begin{aligned} A&=\frac{1}{2}bh \\\\ &=\frac{1}{2}(50)(30) \\\\ &=\frac{1}{2}\times 50 \times 30 \\\\ &=750 \end{aligned}

The total area will be $500+750=1250$

The units will be $m^2.$

24 \ cm

12 \ cm

6 \ cm

18 \ cm

The area is $84 \ cm^2$ and the height is $14 \ cm.$ The calculation we need to do is:

\begin{aligned} A&=\frac{1}{2}bh \\\\ 84&=\frac{1}{2}(b)(14) \\\\ 84&= b\times 7 \\\\ b&=84\div 7 \\\\ b&=12 \end{aligned}

The units will be $cm.$

Area of a triangle GCSE questions

1. A triangle has a base of $6 \ cm$ and perpendicular height $7.2 \ cm.$

Area of a triangle GCSE Question 1

Work out the area of the triangle.

(2 marks)

Show answer

\frac{1}{2}\times 6\times 7.2

(1)

21.6 \ cm^2

(1)

2. The diagram shows a triangular lawn.

Area of a triangle GCSE Question 2

A gardener wants to buy grass seed for this triangular lawn.

A packet of grass seed covers $15 \ m^2$ of ground.

A packets of grass seed costs $£6.99.$

Calculate the cost of buying grass seed to cover the lawn.

(4 marks)

Show answer

\frac{1}{2}\times 9.5\times 8.7=41.325

(1)

41.325\div 15=2.755\text{ or }3

(1)

3\times 6.99

(1)

£20.97

(1)

3. Here is a triangle and a rectangle.

Area of a triangle GCSE Question 3

The area of the rectangle is $5$ times the area of the triangle.

Work out the width $x$ of the rectangle.

(4 marks)

Show answer

\frac{1}{2}\times 6\times 4=12

(1)

12\times 5=60

(1)

60\div 8 = 7.5

(1)

7.5 \ cm

(1)

Learning checklist

You have now learned how to:

Calculate the area of triangles

Derive and apply the formula to calculate and solve problems involving area of triangles

The next lessons are

Beyond GCSE maths – Heron’s formula

The ancient Greeks used a different method for finding the area of a triangle. This is a useful method as we can work out the area of a triangle if we know the side lengths of the triangle. It is based on the semi-perimeter of the triangle (half of the perimeter).

Area of a triangle Beyond GCSE maths image

The semi-perimeter is $s$ and the side lengths of the triangle are $a, b,$ and $c.$

s=\frac{a+b+c}{2}

\text{Area of triangle}=\sqrt{s(s-a)(s-b)(s-c)}

This formula is known as Heron’s formula.

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