One to one maths interventions built for GCSE success

Weekly online one to one GCSE maths revision lessons available in the spring term

Find out more
GCSE Maths Geometry and Measure

Area

Area of an Isosceles Triangle

Area of an Isosceles Triangle

Here we will learn about the area of an isosceles triangle including how to find the area of an isosceles with given lengths and how to calculate those lengths if they are not given.

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 an isosceles triangle?

An isosceles triangle is a type of triangle with two equal sides. The base angles which are opposite to the equal sides are also equal.

Area of an isosceles triangle

There are different types of isosceles triangles:

Isosceles right triangle 

-two equal angles (45° each)

-two equal side lengths

Area of a right isosceles triangle

A special case of an isosceles triangle is an equilateral triangle where all three sides and angles of the triangle are equal. 

What is an isosceles triangle?

What is an isosceles triangle?

How to name a triangle

We can identify a triangle by putting a capital letter on each vertex (corner).

We can then refer to each of the sides of the triangle by using two letters to describe where the line starts and ends.

We can refer to the entire triangle by using all three letters.

E.g.

Name of sides:
side AB, side AC, side BC

Name of triangle:
triangle ABC

Area of an isosceles triangle Image 3

How do we find the area of a triangle?

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

To find the area of a rectangle you must multiply adjacent sides together. 

The area of the rectangle below would be calculated by multiplying the base x height
(b x h).

Area of an isosceles triangle Image 4

We can create an isosceles triangle by drawing two sides from the midpoint of a side of the rectangle to the corners. 

Area of an isosceles triangle Image 5

The rectangle has been split into an isosceles triangle and two congruent (identical) right angled triangles. If we combine the area of the two right angled triangles they will form the pink isosceles triangle.

The area of the isosceles triangle is exactly half the area of the rectangle. 

Area of an isosceles triangle formula

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

Finding the area of an isosceles triangle

This can be shortened to 

\[A=\frac{1}{2} b h\]

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

Your final answer must be given in units2 (cm^{2}, m^{2}, mm^{2}).

How to find the area of an isosceles triangle

In order to find the area of a isosceles triangle:

1Identify the height and base length of your triangle (you might need to calculate these values)

2Write the formula

\[ A=\frac{1}{2} b h\]

3Substitute the values for base and height

4Calculate

How to find the area of an isosceles triangle

How to find the area of an isosceles triangle

Isosceles triangle worksheet

Isosceles triangle worksheet

Isosceles triangle worksheet

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

DOWNLOAD NOW
x
Isosceles triangle worksheet

Isosceles triangle worksheet

Isosceles triangle worksheet

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

DOWNLOAD NOW

Area of isosceles triangles examples

Example 1: given base length and height

Find the area of the triangle below:

Area of an isosceles triangle Example 1 Image 7

  1. Identify the height and base length of the triangle

Area of an isosceles triangle Example 1 Step 1 Image 8

h = 6cm

b = 10cm

2Write down the formula

\[A=\frac{1}{2} b h\]

3Substitute the values for the base and height

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(6)(10) \end{aligned}\]

4Calculate

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(6)(10)\\ &=30cm^2 \end{aligned}\]

Remember: Your final answer must be in units squared.

Example 2: given base length and height

Calculate the area of the triangle below:

Area of an isosceles triangle Example 2 Image 9
Area of an isosceles triangle Example 2 Step 1 Image 10

h = 120cm

b = 4m

Note: You have 2 different units here. You must convert them to a common unit:

120cm = 1.2m

h = 1.2m

b = 4m

\[A=\frac{1}{2} b h\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(4)(1.2) \end{aligned}\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(4)(1.2)\\ &=2.4m^2 \end{aligned}\]

Remember: Your final answer must be in units squared.

Example 3: worded question

Shown below is a triangular shaped field. Each cow needs 2000m^{2} to graze. How many cows can fit into this field?

Area of an isosceles triangle Example 3 Image 11
Area of an isosceles triangle Example 3 Step 1 Image 12

h = 100m

b = 150m

Note: The 125m is not actually needed in this question and is just there to confuse you.

\[A=\frac{1}{2} b h\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(150)(100) \end{aligned}\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(150)(100)\\ &=7500m^2 \end{aligned}\]

Remember: Your final answer must be in units squared.

Now to calculate how many cattle will fit into the field take 7500m^{2} and divide it by 2000m^{2} because each cattle needs that much space to graze on the field.

7500 \div 2000 = 3.75

3 cows will be able to fit in the field.

Example 4: missing height

Find the area of the triangle below:

Area of an isosceles triangle Example 4 Image 13

Area of an isosceles triangle Example 4 Step 1 Image 14

b = 7m

We can find the height by splitting the isosceles triangle into two right triangles. We can then apply Pythagoras’ Theorem to one of the right triangles to calculate its height.

Area of an isosceles triangle Example 4 Step 1.2 Image 15
\[\begin{array}{l} h^{2}=7^{2}-2^{2} \\ h^{2}=49-4 \\ h^{2}=45 \\ h=\sqrt{45} \\ h \approx 6.71 \mathrm{~cm}(2 \mathrm{~d.p.}) \end{array}\]

h = 6.71cm

\[A=\frac{1}{2} b h\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(4)(6.71) \end{aligned}\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(4)(6.71)\\ &=13.42cm^2 \end{aligned}\]

Remember: Your final answer must be in units squared.

Example 5: compound shapes

Below is the floor plan for a new house. Calculate the area of the plan.

Area of an isosceles triangle Example 5 Floor plan
Area of an isosceles triangle Example 5 Floor plan Step 1

Split the plan into 2 shapes. We now have a square and an isosceles triangle.

For the triangle:

b = 8m

h = 11m

\[A=\frac{1}{2} b h\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(8)(11) \end{aligned}\]

\[\begin{aligned} A &=\frac{1}{2} b h \\ &=\frac{1}{2}(8)(11)\\ &=44m^2 \end{aligned}\]

Remember: Your final answer must be in units squared.

Now you must find the area of the square.

Area of square = 8^{2}

                =64m^{2}

Total Area = 44 + 64 = 108m^{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 height or missing length. In order to do this you must rearrange the formula.

To find a missing length given the area:

1Rearrange the formula

2Substitute in the values you know

3Calculate

Step by step guide: Rearranging equations

Example 6: calculating base length

Triangle ABC is an isosceles triangle with an area of 18cm^{2}. The height of the triangle is 6cm. Find the length of the base of the triangle. 

\[A=\frac{1}{2} b h\]

to make b the subject multiply both sides of the formula by 2

\[2A=bh\]

Next we need to divide both sides of the formula by h

\[\frac{2(A)}{h}=b\]

Area = 18cm^{2}

Height = 6cm

\[b=\frac{2(18)}{6}\]
\[\begin{aligned} b&=\frac{2(18)}{6} \\ b&=\frac{36}{6} \\ b&=6 \mathrm{~cm} \end{aligned}\]

Common misconceptions

  • Identifying the correct information to use

A question may give extra information that is not needed to answer it. Carefully identify the relevant pieces of information.

E.g.
To calculate the area here we only need the base and height.

Area of an isosceles triangle Common Misconceptions Image 18

Base= 12cm

Height = 8cm

\[Area = \frac{12 \times 8}{2} = 48cm^{2}\]

We can ignore the values of the other sides (10cm)

  • Units

It is a common error to forget the units for area in the final answer. When calculating area, your answer must always have units squared.

Practice area of an isosceles triangle questions

1. Find the area of the triangle below

 

Area of an isosceles triangle Practice Question 1 Image 19

90cm^{2}
GCSE Quiz False

45cm^{2}
GCSE Quiz True

180cm^{2}
GCSE Quiz False

21cm^{2}
GCSE Quiz False

Using the lengths for base and height, the calculation we must perform is \frac{1}{2} \times 15 \times 6

2. Find the area of the triangle below giving your answer in cm^{2}

 

Area of an isosceles triangle Practice Question 2 Image 20

225cm^{2}
GCSE Quiz False

2250cm^{2}
GCSE Quiz False

22500cm^{2}
GCSE Quiz True

45000cm^{2}
GCSE Quiz False

After converting the height to the appropriate units, the calculation becomes \frac{1}{2} \times 300 \times 150

3. Shown below is a triangular shaped chicken enclosure cage. Each chicken needs 8m^{2} to roam around. How many chickens can fit into this enclosure?

 

Area of an isosceles triangle Practice Question 3 Image 21

125 chickens

GCSE Quiz False

15 chickens

GCSE Quiz True

16 chickens

GCSE Quiz False

30 chickens

GCSE Quiz False

The area of the enclosure is \frac{1}{2} \times 20 \times 12.5 = 125m^{2}. We then consider multiples of 8 to work out how many chickens will fit.

 

Or use a division method: 125 \div 8 = 15.625

 

So 15 chickens will fit.

4. Find the area of the triangle below (give your answer to two decimal places)

 

Area of an isosceles triangle Practice Question 4 Image 22

18.00cm^{2}
GCSE Quiz False

9.00cm^{2}
GCSE Quiz False

8.71cm^{2}
GCSE Quiz True

17.42cm^{2}
GCSE Quiz False

The height of the triangle can be found using Pythagoras’ Theorem.

 

Height = \sqrt{6^{2}-1.5^{2}} Height = 5.809…

 

The area of the triangle is then given by \frac{1}{2} \times 3 \times 5.809…

5. Find the area of the shaded region below (round your answer to one decimal place)

 

Area of an isosceles triangle Practice Question 5 Image 23

24.6m^{2}
GCSE Quiz False

18.3m^{2}
GCSE Quiz True

12.3m^{2}
GCSE Quiz False

19.4m^{2}
GCSE Quiz False

The shape can be split into a rectangle and a triangle.

 

The area of the rectangle is 3 \times 4 = 12m^{2}

 

The area of the triangle is \frac{1}{2} \times 3 \times 4.2 = 6.3m^{2}

 

The total are is 12 + 6.3 = 18.3m^{2}

6. Triangle MNP is an isosceles triangle with an area of 20cm^{2} . The base length of the triangle is 4cm . Find the height of the triangle.

80cm
GCSE Quiz False

5cm
GCSE Quiz False

2.5cm
GCSE Quiz False

10cm
GCSE Quiz True

Starting with the formula and the information we know already,

 

Area = \frac{1}{2}  \times base \times height

20 = \frac{1}{2} \times 4  \times height

20 = 2  \times height

 

So the height is 10cm

Area of an isosceles triangle GCSE questions

1. This pattern is made from three identical isosceles triangles. Find the total area.

 

Area of an isosceles triangle GCSE Question 1 Image 24

 

(3 marks)

Show answer
10 \div 2 = 5

(1)

 

One triangle:

\begin{aligned} A&= \frac{1}{2} \times 5 \times 12\\ A&=30cm^{2} \end{aligned}

(1)

 

Total area:

3 \times 30 = 90cm^{2}

(1)

2. (a) Find the area of the following triangle:

 

Area of an isosceles triangle GCSE Question 2a Image 25

(b) The triangle below has the same area as the triangle in
part (a).

 

Area of an isosceles triangle GCSE Question 2b Image 26

Work out the height of this triangle.

(3 marks)

Show answer
\begin{aligned} A&= \frac{1}{2} \times 8 \times 12\\ A&=48cm^{2} \end{aligned}

(1)

Rearrange area of triangle:

\begin{aligned} A &= \frac{1}{2} bh \\ 2A &= bh\\ h & = \frac{2A}{b} \end{aligned}

 

Substitute in values:

h= \frac{2 \times 48}{16}

(1)

\begin{array}{l} h= \frac{96}{16}\\ h=6cm \end{array}

(1)

3. Calculate the area of the triangle.

 

Area of an isosceles triangle GCSE Question 3 Image 27

 

(4 marks)

Show answer

Find height of triangle using Pythagoras:

h^{2}=5^{2}-3^{2}

(1)

\begin{aligned} h^{2}&=16\\ h&=4 \end{aligned}

(1)

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

(1)

A=12cm^{2}

(1)

Learning checklist

You have now learned how to:

  •  Apply formula to calculate and solve problems involving the area of triangles
  •  Use Pythagoras’ Theorem to solve problems involving triangles

The next lessons are

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.

Find out more about our GCSE maths revision programme.