Area of a rhombus

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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.

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Area of a rhombus worksheet

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

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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

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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

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