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.

There are different types of isosceles triangles:

Isosceles right triangle

-two equal angles (45° each)

-two equal side lengths

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

### 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

### 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).

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

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

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

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.

## 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

## Area of isosceles triangles examples

### Example 1: given base length and height

Find the area of the triangle below:

1. Identify the height and base length of the triangle

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}

### Example 2: given base length and height

Calculate the area of the triangle below:

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}

### 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?

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}

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:

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.

$\begin{array}{l} h^{2}=7^{2}-2^{2} \\ h^{2}=49-4 \\ h^{2}=45 \\ h=\sqrt{45} \\ h \approx 6.71 \mathrm{~cm}(2 \mathrm{~d.p.}) \end{array}$

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}

### Example 5: compound shapes

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

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}

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.

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

90cm^{2}

45cm^{2}

180cm^{2}

21cm^{2}

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}

225cm^{2}

2250cm^{2}

22500cm^{2}

45000cm^{2}

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?

125 chickens

15 chickens

16 chickens

30 chickens

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)

18.00cm^{2}

9.00cm^{2}

8.71cm^{2}

17.42cm^{2}

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…

24.6m^{2}

18.3m^{2}

12.3m^{2}

19.4m^{2}

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

5cm

2.5cm

10cm

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.

(3 marks)

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:

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

Work out the height of this triangle.

(3 marks)

\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.

(4 marks)

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

## 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.