Area of an isosceles triangle

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GCSE Maths Geometry and Measure Area

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.

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

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

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

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 ( $cm^{2}, m^{2}, mm^{2}$ ).

To find the area of an isosceles triangle when only the side lengths have been given, we will need to calculate the height of the triangle using Pythagoras’ Theorem

Finding the height using Pythagoras’ Theorem

Let the base of the isosceles triangle be $b$ and the two equal sides be $l.$

Drawing a perpendicular line from the top vertex to the base forms the height, $h,$ and also forms a right angled triangle with hypotenuse $l$ and short sides $h$ and $\cfrac{1}{2} \, b.$

Area of an Isosceles Triangle image 1 Uk

h=\sqrt{{{l}^{2}}-{{\left( \cfrac{1}{2} \, b \right)}^{2}}}

We can also find the area of the isosceles triangle using the area of a triangle formula

Area=\cfrac{1}{2} \, ab\sin C

If we know (or can find) the two equal sides, $l$ and the angle between them, $\theta ,$ the formula would give the area as

Area of an Isosceles Triangle image 2 UK

A=\cfrac{1}{2} \, {{l}^{2}}\sin \left( \theta \right)

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 if given. (You might need to calculate these values)

2Write the appropriate formula
$A=\cfrac{1}{2} \,b h$ or $A=\frac{1}{2}{{l}^{2}}\sin \left( \theta \right)$

3Substitute the values into the formula

4Calculate

How to find the area of an isosceles triangle

Isosceles triangle worksheet

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

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Isosceles triangle worksheet

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

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Related lessons on area

Area of isosceles triangles 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 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

Identify the height and base length of the triangle if given.

Area of an isosceles triangle Example 1 Step 1 Image 8

h = 6cm

b = 10cm

2Write down the appropriate formula

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

3Substitute the values into the formula

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

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

Identify the height and base length of the triangle if given.

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

Write down the appropriate formula

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

Substitute the values into the formula.

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

Calculate

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

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

Identify the height and base length of the triangle if given.

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.

Write down the appropriate formula

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

Substitute the values into the formula.

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

Calculate

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

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

Identify the height and base length of the triangle if given.

Area of an isosceles triangle Example 4 Step 1 Image 14

b = 4cm

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

Write down the appropriate formula

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

Substitute the values into the formula.

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

Calculate

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

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

Identify the height and base length of the triangle if given.

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

Write down the appropriate formula

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

Substitute the values into the formula.

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

Calculate

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

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 = $44 + 64 = 108m^{2}$

Example 6: Using the ½ab sin C formula

Below shows an isosceles triangle with two sides of length $10 \, cm$ and a base angle of $75^{\circ}.$

Area of an Isosceles Triangle image 3 uk

Find the area.

Identify the height and base length of your triangle if given. (You might need to calculate these values)

We have the length of the two equal sides, $10\, cm$ and the base angle of $75^{\circ}.$

To use the formula $A=\frac{1}{2}{{l}^{2}}\sin \left( \theta \right)$ we need the angle between the equal sides.

The base angles are both $75^{\circ},$ so $180^{\circ}-150^{\circ}=30^{\circ}$ for the remaining angle

Area of an Isosceles Triangle image 4 uk

Write down the appropriate formula

A=\cfrac{1}{2} \, {{l}^{2}}\sin \left( \theta \right)

Substitute the values into the formula.

A=\cfrac{1}{2} \, {{(10)}^{2}}\sin \left( 30 \right)

Calculate

\begin{aligned} A&=\cfrac{1}{2} \, {{\left( 10 \right)}^{2}}\sin \left( 30 \right) \\\\ & =25 \, cm^2 \end{aligned}

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 7: 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.

Rearrange the formula

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

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 $h$

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

Now substitute the given values.

Area = $18cm^{2}$

Height = $6cm$

\[b=\frac{2(18)}{6}\]

Calculate

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

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

90cm^{2}

45cm^{2}

180cm^{2}

21cm^{2}

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

225cm^{2}

2250cm^{2}

22500cm^{2}

45000cm^{2}

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

$125$ 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.

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\dots

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 area is $12 + 6.3 = 18.3m^{2}$

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.

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 isosceles triangle:

Area of an isosceles triangle GCSE Question 2a Image 25

(b) The isosceles 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 isosceles 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

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