Area of a rhombus

GCSE Tutoring Programme

"Our chosen students improved 1.19 of a grade on average - 0.45 more than those who didn't have the tutoring."

Teacher-trusted tutoring

In order to access this I need to be confident with:

Arithmetic Types of quadrilaterals Area of a rectangle Area of a triangle Pythagorean theorem Squares and square roots

This topic is relevant for:

GCSE Maths Geometry and Measure Area

Area Of A Rhombus

Here we will learn about finding the area of a rhombus, including compound area questions, questions with missing side lengths and questions involving unit conversion.

There are also area of a rhombus 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 rhombus?

The area of a rhombus is the amount of space inside the rhombus. It is measured in units squared ( $cm^2, \ m^2, \ mm^2$ etc).

Area formula:

A=\frac{1}{2}\times{D}\times{d}

Where $D$ is the long diagonal of the rhombus (the base) and $d$ is the short diagonal in the rhombus (the height of the rhombus).

A rhombus is a polygon and a quadrilateral ( $4$ sided shape) where opposite sides are parallel, all sides are equal length and opposite angles are equal. The interior angles of a rhombus add up to $360^{\circ}$ . A rhombus is a special type of a parallelogram.

The diagonals of a parallelogram bisect each other at a right angle as shown below:

To calculate the area of a rhombus, we need to know the length of each diagonal of the rhombus.

If we move the two triangles from half of the rhombus to the opposing side, we get a rectangle:

The width of the rectangle is half of the long diagonal of the rhombus ( $D$ ), and the height of the rectangle is the length of the short diagonal in the rhombus ( $d$ ).

This gives the area of the rhombus formula:

A=\frac{1}{2}\times{D}\times{d}

What is the area of a rhombus?

How to find the area of a rhombus

In order to find the area of a rhombus:

Identify the length of the diagonals.
Write down the formula for the area of a rhombus.
Substitute the given values of the diagonals and solve.
Write down your final answer, including the units.

How to find the area of a rhombus

Area of a rhombus worksheet

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

DOWNLOAD FREE

Area of a rhombus worksheet

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

DOWNLOAD FREE

Related lessons on area

Area of a rhombus 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 rhombus examples

Example 1: finding area given the length of the diagonals

Find the area of the rhombus below:

Identify the length of the diagonals.

\begin{aligned} &AC = 2m \\\\ &BD = 9m \end{aligned}

These are the diagonals of a rhombus which correspond to the base and height of the corresponding rectangle.

2Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

3Substitute the given values of the diagonals and solve.

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{9}\times{2}\\\\ &=9 \end{aligned}

4Write down your final answer, including the units.

In this case we are working with metres so our final answer must be in square metres.

A=9m^2

Example 2: finding the area of a rhombus requiring converting units

Calculate the area of the rhombus below:

Identify the length of the diagonals.

\begin{aligned} &MO = 50cm \\\\ &NP= 1m \end{aligned}

We have two different units for the length and width so we must change the measures to a common unit.

In this case, let us change both units to centimetres.

As $1m=100cm$ .

$\begin{aligned} &MO = 50cm \\\\ &NP = 100cm \end{aligned}$

Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

Substitute the given values of the diagonals and solve.

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{100}\times{50}\\\\ &=2500 \end{aligned}

Write down your final answer, including the units.

As the area has been calculated using the lengths in centimetres, the area unit is square centimetres.

$A=2500\text{ cm}^2$

Example 3: worded problem

A rhombus $ABCD$ has the point $O$ as the point of intersection of the two diagonals $AC$ and $BD$ . The length $OB=3cm$ , and $OC=5cm$ . Calculate the area of the rhombus.

Identify the length of the diagonals.

As the point $O$ is the intersection of the two diagonals, the lengths $OB$ and $OC$ are half of the length of each diagonal. This means that as $OB=3cm, \ AB=3 \times 2=6cm$ and as $OC=5cm, \ CD=5 \times 2=10cm.$

Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

Substitute the given values of the diagonals and solve.

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{10}\times{6}\\\\ &=30 \end{aligned}

Write down your final answer, including the units.

As the diagonal lengths are given in centimetres, the area unit is square centimetres.

$A=30\text{ cm}^2$

Example 4: calculating the diagonal given area and the other diagonal

Calculate the length of $AC$ given the length of $BD=14m$ and the area of the rhombus is $35m^2$ . Use the diagram below to help you.

Identify the length of the diagonals.

\begin{aligned} &BD = 14m \\\\ &AC = x \end{aligned}

In this case we are also given Area $= 35m^2$ .

Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

Substitute the given values of the diagonals and solve.

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d} \\\\ 35&=\frac{1}{2}\times{14}\times{x} \\\\ 35&=7x \\\\ x&=5 \end{aligned}

Write down your final answer, including the units.

In this case we are asked to find the length of a diagonal and not area so our answer should be written in metres:

$AC=5\text{ m}$

Example 5: area given side length and one diagonal

$ABCD$ is a rhombus where the edges $AB$ and $CD$ are parallel. Given that $AB=20cm$ and $BD=32cm,$ calculate the area of the rhombus.

Identify the length of the diagonals.

\begin{aligned} &BD = 32cm \\\\ &AC = x \end{aligned}

We need to calculate the length of $AC.$ In this question we are given one side of the rhombus which is $AB=20cm$ .

Note: Since we are given the length of the side $AB$ we now know the length of any side of the rhombus since all sides of a rhombus are the same length.

We can split the rhombus into $4$ right-angled triangles. If we take triangle $OAB$ , the hypotenuse $AB=20cm$ and $OB=16cm$ since it is half the length of $BD$ . We now need to calculate the length of $OA$ .

Area of a Rhombus Example 5 Step 1 Image 1

As $OAB$ is a right-angled triangle, we can use Pythagoras’ theorem to calculate $OA$ .

Area of a Rhombus Example 5 Step 1 Image 2

$\begin{aligned} c^{2}&=a^{2}+b^{2}\\\\ 20^{2}&=16^{2}+x^{2}\\\\ 400&=256+x^{2}\\\\ x^{2}&=144\\\\ x&=\sqrt{144}\\\\ x&=12 \end{aligned}$

As $x=OA=12cm, \ AC=12 \times 2=24cm$

Now that we have the length of two of the diagonals of the rhombus, we can calculate the area of the rhombus $ABCD$ .

Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

Substitute the given values of the diagonals and solve.

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{32}\times{24}\\\\ &=384 \end{aligned}

Write down your final answer, including the units.

The area of the rhombus $ABCD$ is $384cm^2$ .

Example 6: calculate the area using trigonometry (higher only)

A rhombus has a side length of $13cm$ . One pair of interior angles is double the other pair of interior angles. Calculate the area of the rhombus to $2$ decimal places.

Identify the length of the diagonals.

Sketching a diagram first, we have:

Area of a Rhombus Example 6 Step 1 Image 1

As the sum of angles in a rhombus is $360^{\circ}$ , we can quickly calculate the value of $x$ by forming and solving an equation:

$\begin{aligned} 2x+x+2x+x&=360 \\\\ 6x&=360 \\\\ x&=60^{\circ} \end{aligned}$

So the angles in the rhombus are $60^{\circ}$ and $120^{\circ}$ . If we filled these in on the diagram, we have:

Area of a Rhombus Example 6 Step 1 Image 2

We can calculate the length $AC$ by using the cosine rule $c^{2}=a^{2}+b^{2}-2ab\cos(\theta)$ , looking at the top half of the rhombus, we have:

Area of a Rhombus Example 6 Step 1 Image 3

Labelling the sides and angles according to the formula ( $\theta$ must be the included angle between the two sides $a$ and $b$ , we have:

Area of a Rhombus Example 6 Step 1 Image 4

Substituting the values of $a, b,$ and $\theta$ into the cosine rule, we have:

$\begin{aligned} c^{2}&=a^{2}+b^{2}-2ab\cos(\theta) \\\\ c^{2}&=13^{2}+13^{2}-2\times{13}\times{13}\times\cos(120)\\\\ c^{2}&=507\\\\ c&=13\sqrt{3} \end{aligned}$

So the long diagonal $AC=13\sqrt{3}\text{ cm}$

Repeating the same process for the diagonal BD, we have:

Area of a Rhombus Example 6 Step 1 Image 5

Substituting the values of $a, b,$ and $\theta$ into the cosine rule, we have:

$\begin{aligned} c^{2}&=a^{2}+b^{2}-2ab\cos(\theta) \\\\ c^{2}&=13^{2}+13^{2}-2\times{13}\times{13}\times\cos(60) \\\\ c^{2}&=169 \\\\ c&=13 \end{aligned}$

So the short diagonal $BD=13\text{ cm}$ .

(For this example, BCD is an equilateral triangle!)

Write down the formula for the area of a rhombus.

A=\frac{1}{2}\times{D}\times{d}

Substitute the given values of the diagonals and solve.

As the long diagonal $AC=13\sqrt{3}\text{ cm}$ and the short diagonal $BD=13\text{ cm}$ , we have:

$\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{13\sqrt{3}}\times{13}\\\\ &=146.3582932…\\\\ &=146.36\text{ cm}^2\text{ (2dp)} \end{aligned}$

Write down your final answer, including the units.

The area of the rhombus $ABCD$ is $146.36cm^2 \ (2dp).$ .

Common misconceptions

Using incorrect units for the answer

A common error is to forget to include squared units when asked to calculate area.

Forgetting to convert measures to a common unit

Before using the formula for calculating the area of a rectangle we need to ensure that units are the same. If different units are given (E.g. length $= 4m$ and width $= 3cm$ ) then you must convert them either both to $cm$ or both to $m.$

Using half of the diagonal

As the area of a rhombus is half the length of one diagonal multiplied by the length of the other diagonal, the other diagonal is also halved within the calculation, leaving a solution that is out by a factor of $2.$

Calculating the area of a square (using side lengths)

The area of a rhombus is calculated by using the side lengths, instead of the lengths of the diagonals.

Practice area of a rhombus questions

24m^2

48m^2

16m^2

48m

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{12}\times{4}\\\\ &=24 \end{aligned}

A=24\text{ m}^2

11250cm^2

28125cm^2

56.25cm^2

56250cm^2

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{250}\times{450}\\\\ &=56250 \end{aligned}

A=56250\text{ cm}^2

24

48

96

192

Calculating the area of the rhombus, we have

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{12}\times{16}\\\\ &=96 \end{aligned}

A=96\text{ m}^2

As the field is $96m^2$ and each chicken requires $4m^2, \ 96\div{4}=24$ chickens

9m

18m

54m

324m

As $A=54m^2$ and $AP=3m, \ AC=32=6m.$

Using the formula for the area of a rhombus, we have

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ 54&=\frac{1}{2}\times{6}\times{x}\\\\ 54&=3x\\\\ x&=18 \end{aligned}

x=18\text{ m}

24m^2

48m^2

6m^2

20m^2

We can calculate half the length of $AC$ by using Pythagoras’ theorem. Let $M$ be the centre of the rhombus. The distance $BM$ is half the distance $BD$ and so $BM=4m.$

Area of a Rhombus Practice Question 5 Explanation Image

Using Pythagoras’ theorem to calculate the length of $AM$ , we have:

\begin{aligned} AM^{2}&=5^{2}-4^{2}\\\\ &=25-16\\\\ &=9\\\\ AM&=3 \end{aligned}

As $AM=3m, \ AC=3 \times 2=6m.$

The area of the rhombus is therefore:

\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{6}\times{8}\\\\ &=24 \end{aligned}

A=24\text{ m}^2

19.2cm^2

12.5cm^2

25.0cm^2

20.0cm^2

First we need to calculate the value of the two pairs of angles in the rhombus:

\begin{aligned} x+3x-20+x+3x-20&=360\\\\ 8x-40&=360\\\\ 8x&=400\\\\ x&=50 \end{aligned}

x=50^{\circ}

3x-20=130^{\circ}

So we have a rhombus with a side length of $5cm$ and the interior angles are $50^{\circ}$ and $130^{\circ}.$

Area of a Rhombus Practice Question 6

Calculating the distance $PR$ using the cosine rule, we have:

PR=\sqrt{5^{2}+5^{2}-2\times{5}\times{5}\times\cos(130)}=9.06307787

Calculating the distance $QS$ using the cosine rule, we have:

QS=\sqrt{5^{2}+5^{2}-2\times{5}\times{5}\times\cos(50)}=4.226182617

The area of the rhombus is therefore:
$\begin{aligned} A&=\frac{1}{2}\times{D}\times{d}\\\\ &=\frac{1}{2}\times{9.06307787}\times{4.226182617}\\\\ &=19.15111108 \end{aligned}$

A=19.2\text{ cm}^2\text{ (1dp)}

Area of a rhombus GCSE questions

1. Shown below is a rhombus where $AM=6m$ , and $BM=10m$ . Calculate the area of the rhombus.

Area of a Rhombus GCSE Question 1

(3 marks)

Show answer

AC=12m, \ BD=20m

(1)

\frac{12 \times 20}{2}

(1)

120m^2

(1)

2. (a) A rectangular logo is composed of $8$ congruent rhombuses. Calculate the area of one rhombus.

Area of a Rhombus GCSE Question 2a

(b) Calculate the perimeter of the logo. Write your answer as a surd in its simplest form.

(6 marks)

Show answer

(a)

12\div{3}=4\text{m and }4\div{2}=2\text{m}

(1)

\frac{4 \times 2}{2}

(1)

4\text{ m}^{2}

(1)

(b)

2^{2}+1^{2}=5

(1)

Side length of a rhombus $=\sqrt{5}$

(1)

16\sqrt{5}

(1)

3. Ms. Polly is looking to cover her garden with new grass. One roll of grass covers $6$ square metres and costs $£9.$ How much would it cost to cover the entire ground with grass?

Area of a Rhombus GCSE Question 3

(5 marks)

Show answer

\sqrt{25^{2}-7^{2}}

(1)

24m

(1)

\frac{48 \times 14}{2}

(1)

(336 \div 6) \times 9

(1)

£504

(1)

Learning checklist

You have now learned how to:

Calculate the area of a rhombus and related composite shapes

Calculate the area of parallelograms

Recognise when it is possible to use formulae for area

Derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles and parallelograms

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 tuition programme.

Introduction

What is the area of a rhombus?

How to find the area of a rhombus

Area of a rhombus worksheets

Area of a rhombus examples

↓

Example 1: finding area given the length of the diagonals Example 2: finding the area of a rhombus requiring converting units Example 3: worded problem Example 4: calculating the diagonal given area and the other diagonal Example 5: area given side length and one diagonal Example 6: calculate the area using trigonometry (higher only)

Common misconceptions

Practice area of a rhombus questions

Area of a rhombus GCSE questions

Learning checklist

Next lessons

Still stuck?