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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). 

Area of a Rhombus Image 1

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.

Area of a Rhombus Image 2

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

Area of a Rhombus Image 3

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:

Area of a Rhombus Image 4

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?

What is the area of a rhombus?

How to find the area of a rhombus

In order to find the area of a rhombus:

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

How to find the area of a rhombus

How to find the area of a rhombus

Area of a rhombus worksheet

Area of a rhombus worksheet

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

Area of a rhombus worksheet

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:

Area of a Rhombus Example 1

  1. 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:

Area of a Rhombus Example 2

\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}

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

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

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 rhombic prism has a cross-section ABCD . The point O is the point of intersection of the two diagonals AC and BD . The length OB=3cm , and OC=5cm . Given that the depth of the prism is 4cm, calculate the volume of the prism.

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.

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

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

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.

Area of a Rhombus Example 4
`

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


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

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

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

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.

\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 .

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

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

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.

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!)

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

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}

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

1. Find the area of the rhombus below:

 

Area of a Rhombus Practice Question 1

24m^2
GCSE Quiz True

48m^2
GCSE Quiz False

16m^2
GCSE Quiz False

48m
GCSE Quiz False
\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

2. The rhombus MNOP has the following information:

 

    MP=250cm

 

    NO=4.5m

 

Calculate the area of the rhombus in square centimetres.

11250cm^2
GCSE Quiz False

28125cm^2
GCSE Quiz False

56.25cm^2
GCSE Quiz False

56250cm^2
GCSE Quiz True
\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

3. Mr. Perry owns a farm and is looking to make the chicken enclosure shown below. Each chicken needs 4 square metres to move around freely. What is the maximum number of chickens that can fit into the enclosure?

 

Area of a Rhombus Practice Question 3

24
GCSE Quiz True

48
GCSE Quiz False

96
GCSE Quiz False

192
GCSE Quiz False

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

4. The area of rhombus ABCD is 54m^2. The centre of the rhombus is located at point P . Given that AP=3m , calculate the length BD.

 

Area of a Rhombus Practice Question 4

9m
GCSE Quiz False

18m
GCSE Quiz True

54m
GCSE Quiz False

324m
GCSE Quiz False

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{ cm}

5. In the rhombus below AB = 5m, \ BD = 8m. Calculate the area of the rhombus.

 

Area of a Rhombus Practice Question 5

24m^2
GCSE Quiz True

48m^2
GCSE Quiz False

6m^2
GCSE Quiz False

20m^2
GCSE Quiz False

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=3cm, \ AC=3 \times 2=6cm.

 

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{ cm}^2

6. (Higher only) A rhombus PQRS has side length 5cm . The rhombus has two pairs of interior angles, equal to x and 3x-20 .

 

Use the cosine rule c^2=a^2+b^2-2ab \cos(\theta) to calculate the area of the rhombus to 1 decimal place.

19.2cm^2
GCSE Quiz True

12.5cm^2
GCSE Quiz False

25.0cm^2
GCSE Quiz False

20.0cm^2
GCSE Quiz False

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

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