# Area of a parallelogram (2023)

Here we will learn how to find the area of a parallelogram, including compound area questions, questions with missing side lengths and questions involving unit conversion.

There are also area of a area of a parallelogram 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 parallelogram?

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

A parallelogram is a quadrilateral ( 4 sided shape) with 2 pairs of parallel sides. The parallel sides are the same length and the opposite angles are equal. The interior angles of the parallelogram add up to 360° .

The area of a parallelogram is calculated by multiplying the base of a parallelogram by its perpendicular height.

Area of a parallelogram formula:

\begin{aligned}\text { Area }_{\text {parallelogram }}&=\text { base } \times \text { perpendicular height } \\\text { Area }_{\text {parallelogram }}&=b \times h\end{aligned}

Let’s see why this works:

If we cut off the triangle at one end of a parallelogram and attach it to the other we form a rectangle. The formula for the area of a rectangle is length\times width (or in this case base \times height).

Our final answer must be in units squared
E.g. square centimetres ( cm^{2}) , square metres ( m^{2} ), square feet ( ft^{2} ) etc.

### What is the area of a parallelogram? ## Parallelogram

A parallelogram is a 2D shape with opposite sides that are both parallel and equal in length. The diagonals of a parallelogram bisect each other. We can categorise them as quadrilaterals because they have 4 sides.

We need to be able to identify the three special types of parallelograms.

E.g.

A parallelogram where all sides are equal is a rhombus.

E.g.

A parallelogram in which all angles are right angles, and the diagonals are equal is a rectangle.

E.g.

A parallelogram with all equal sides and all angles equal to 90 degree, and equal diagonals is a square.

A parallelogram is, essentially, a slanted rectangle, so we can use the same calculation to find the area of a parallelogram as you would calculate the area of a rectangle, as long as you use the perpendicular height rather than the slant length.

E.g.

Calculate the area of the parallelogram.

Area = 10 x 6 = 60

Let’s explore more about parallelograms below.

## How to find the area of a parallelogram

In order to find the area of a parallelogram:

1. Identify the base and the perpendicular height of the parallelogram.
2. Write down the formula for the area of a parallelogram.
3. Substitute the given values and calculate.

### How to find the area of a parallelogram  ### Area of a parallelogram worksheet Get your free Area of a parallelogram worksheet of 20+ questions and answers. Includes reasoning and applied questions.

x ### Area of a parallelogram worksheet Get your free Area of a parallelogram worksheet of 20+ questions and answers. Includes reasoning and applied questions.

## Related lessons on area

Area of a parallelogram 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
• Area of a circle
• Area of a trapezium
• Area of an equilateral triangle
• Area of compound shapes
• Pi r squared
• Area of a rhombus
• Area of a right angled triangle
• Area of an isosceles triangle
• Area of a triangle
• How to work out area
• Area of a rectangle

## Area of a parallelogram examples

### Example 1: finding area given the base and perpendicular height

Find the area of the parallelogram below:

1. Identify the base and perpendicular height of the parallelogram.

Base = 10m

Perpendicular height = 6m

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

\text { Area }_{ \text { parallelogram }}=\text { Base } \times \text { Perpendicular Height }

3Substitute the given values and calculate.

\begin{aligned}\text { Area }_{\text {parallelogram }}&=\text { Base } \times \text { Perpendicular Height } \\&=10 \times 6 \\&=60\end{aligned}

\text { Area }_{\text {parallelogram }}=60 \mathrm{~m}^{2}

### Example 2: finding the area of a parallelogram requiring converting units

Find the area of the parallelogram below:

We have two different units for the base and perpendicular height so we need to change the measures to a common unit.

Let’s convert both measures to metres.

There are 100cm in a metre, so if we divide the base by 100 we get 8m

Base = 8m

Perpendicular Height = 6m

NOTE: It is important to note that the 6.3m is the length of the diagonal and not the perpendicular height.

\text { Area }_{ \text { parallelogram }}=\text { Base } \times \text { Perpendicular Height }

(Video) Area of Parallelograms | How to Find the Area of a Parallelogram

\begin{aligned}\text { Area }_{ \text { parallelogram }}&=\text { Base } \times \text { Perpendicular Height }\\&=8\times6\\&=48\\end{aligned}

$\text { Area }_{ \text { parallelogram }}=48m^{2}$

### Example 3: worded question

A painter is painting a logo on the side of an office block. The logo is composed of four identical parallelograms and is shown below. Each tin of paint costs £1.20 and contains enough paint to cover an area of 2.5m^{2} . How much will it cost to paint the logo?

Base = 4m

Perpendicular height = 2m ( 8\div4 parallelograms)

\text { Area }_{ \text { parallelogram }}=\text { Base } \times \text { Perpendicular Height }

Since there are 4 identical parallelograms in the logo we need to multiply the area by 4 .

\text { Area }_{\text {Logo }}=4 \times(\text { Base } \times \text { Perpendicular Height })

\begin{aligned}\text { Area }_{\text {Logo }} &=4 \times(\text { Base } \times \text { Perpendicular Height }) \\&=4 \times(4 \times 2) \\&=32\end{aligned}

\text { Area }_{\text {Logo }}=32 \mathrm{~m}^{2}

Now that we have calculated the area we need to find how many tins of paint are needed to paint the logo.
Since each paint tin covers an area of 2.5m^{2} we need to divide 2.5 into 32 .

32 \div 2.5=12.8 tins

Since you cannot buy 0.8 of a tin we need to round the number of tins to 13 .

Each can costs £1.20 so 13 \times 1.2=£ 15.60

Therefore the total cost of painting the logo is £15.60 .

### Example 4: calculating base length given the area

Find the base length of the parallelogram below:

Base = ?

Perpendicular height = 11cm

In this question we are given the area of the parallelogram as 242cm^{2} , which we can use to calculate the base.

\text { Area }_{ \text { parallelogram }}=\text { Base } \times \text { Perpendicular Height }

\begin{aligned}\text { Area }_{ \text { parallelogram }} &=\text { Base } \times \text { perpendicular Height } \\242 &=\text { Base } \times 11 \\\text { Base } &=\frac{242}{11} \\\text { Base } &=22\end{aligned}

Base length = 22cm

### Example 5: compound area

Calculate the area of the shaded region below:

Before finding the area of the shaded region, we need to split the shape into two parallelograms. We can cut the shape in half because both parallelograms are identical.

(Video) Areas of Parallelograms

Each parallelogram has a perpendicular height of 5cm and base length of 8cm .

Base = 8cm

Perpendicular height = 5cm

Note: We will also need to subtract the area of the square since it is not shaded.

\text { Area }_{ \text { parallelogram }}=\text { Base } \times \text { Perpendicular Height }

There are two identical parallelograms so we need to multiply the area of the parallelogram by 2 and then subtract the area of the square which is calculated by 3^{2} :

\text { Area }_{\text {Shaded Region }}=2 \times(\text { Base } \times \text { Perpendicular Height })-3^{2}

\begin{aligned}\text { Area }_{\text {Shaded Region }} &=2 \times(\text { Base } \times \text { Perpendicular Height })-s^{2} \\&=2 \times(8 \times 5)-3^{2} \\&=71\end{aligned}

\text { Area }_{\text {Shaded Region }}=71 \mathrm{~cm}^{2}

### Example 6: finding the area involving algebra question

Calculate the value of x in the parallelogram below:

Base = 2x + 3

Perpendicular height = 2cm

We are also given the area:

Area = 14cm^{2}

\text { Area }_{ \text { parallelogram }}=\text { Base } \times \text { Perpendicular Height }

\text { Area }_{\text {Parallelogram }}=\text { Base } \times \text { Perpendicular Height }

\begin{aligned}&14=2(2 x+3) \\&14=4 x+6 \\&4 x=8 \\&x=2\end{aligned}

x=2

### 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 parallelogram, we need to ensure the units used are the same. If different units are given (E.g. length = 4m and width = 3cm ) we must convert them both to either cm or both to m .

• Length of the slanted side and perpendicular height

Sometimes in a question we are given additional lengths which are not needed in our calculations. An example of this is being given the length of one of the slanted sides of the parallelogram. To calculate the area we must ignore the slant length and instead use the vertical height.

### Practice Area of a parallelogram questions

1. Find the area of the parallelogram below: 17.5m 66m^{2} 17.5m 66m b \times h = 12 \times 5.5 = 66m^{2}. Remember to have units squared in your final answer.

2. Find the area of the parallelogram below: 31.5m^{2} (Video) Area of parallelograms intuition | Algebra I | High School Math | Khan Academy

3150m^{2} 22.4m^{2} 14.4m^{2} Convert 450cm into metres by dividing by 100 . Multiply the base length (7m) by the converted perpendicular height (4.5m) to equal 31.5m^{2} . Be careful not to multiply by the length of the diagonal instead of the perpendicular height.

3. Calculate the area of the garden below: 25.2m^{2} 12.2m^{2} 31.2m^{2} 37.2m^{2} Find the area of the parallelogram by multiplying the base length and the height. Find the area of the triangle by multiplying the base and the height and dividing by 2 . Add together the area of the triangle and parallelogram.

4. Calculate the length of the base of the parallelogram below: 22cm 2662cm^{2} 22cm^{2} 2662cm Divide the area by the perpendicular height of 11m . Remember that you are not calculating the area so your final answer should not be in units squared.

5. Calculate the area of the shaded region below: 126cm^{2} 224cm^{2} 98cm^{2} 126.98cm^{2} Convert all measures to cm . Find the area of the parallelogram first by multiplying the base times the height. Add on the area of the rectangle which is calculated by multiplying the length and the width.

6. Calculate the base length in the parallelogram below: 4cm

(Video) Area of a Parallelogram | MathHelp.com 5cm 16cm 33cm Multiply the base times the perpendicular height, Remember to multiply the two by both terms. Rearrange the equation to calculate x . Substitute the value of x back into 4x-11 to calculate the base length.

### Area of a parallelogram GCSE questions

1. Find the area of the parallelogram. Remember to show your full working. (2 marks)

38\times19

(1)

722m^{2}

(1)

2. A landscaper is planning a rectangular garden for a client. The garden will feature a patio and sandpit both in the shape of parallelograms. The remainder of the garden will be covered in grass. Below is a blueprint of the garden.
(a)What area of the garden will be covered in grass? (b) A roll of grass covers an area of 3m^{2} and costs £14.50 . How much will it cost to put grass in the garden?

(7 marks)

(a)
Area of rectangle = 15\times7 = 105

(1)

Area of both parallelograms: 5\times4 = 20 and 4.5\times1.5 = 6.75

(1)

Area of rectangle – area of parallelograms= 105 – 20 – 6.75

(1)

78.25m^{2}

(1)

(b)
78.25\div3 = 26.08 rolls of grass

(1)

27\times14.5

(1)

£391.50

(1)

3. Calculate the value of x in the diagram below: (4 marks)

6 \times (3x-9)

(1)

18x-54=36

(1)

Evidence of rearranging equation to solve for x

(1)

x = 5

(1)

## Learning checklist

You have now learned how to:

• Calculate and compare the area of parallelograms using standard units
• Calculate the area of parallelograms and related composite shapes

## The next lessons are

• How to work out perimeter
• Symmetry
• Circles, sectors and arcs

## 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 revision programme.

(Video) Area of a Parallelogram (Introduction)

## FAQs

### How do you find the area of parallelograms? ›

The formula for the area of a parallelogram is base times height, just like the formula for the area of a rectangle.

How do you find the area of a 4 sided parallelogram? ›

The parallelogram is a quadrilateral with opposite sides parallel; it always has four sides, and one longer side will always be its base. To find its area, you need to know its height. Multiply the length of the base × height, and express the answer in square units.

What is parallelogram and its area? ›

A parallelogram is a two-dimensional geometrical shape whose sides are parallel to each other. It is a type of polygon having four sides (also called quadrilateral), where the pair of parallel sides are equal in length. The Sum of adjacent angles of a parallelogram is equal to 180 degrees.

Is the area of a parallelogram the same as a rectangle? ›

Because the parallelogram and rectangle are composed of the same parts, they necessarily have the same area. (See the definition of area for more about why those areas are the same.)

What is the formula to find the area? ›

The area is measurement of the surface of a shape. To find the area of a rectangle or a square you need to multiply the length and the width of a rectangle or a square. Area, A, is x times y.

How do you find the area of a ABCD parallelogram? ›

∴ Area of parallelogram = Height × Base of parallelogram =6×(3+4)=6×7=42 cm2.

What is a parallelogram with 4 sides? ›

A rhombus is a parallelogram with all four sides congruent to each other. diamond-like shape. A square is a parallelogram with four congruent sides and four right angles. In other words, a square is a rectangle and a rhombus.

What is a parallelogram with 4 sides of equal length? ›

A rhombus is a quadrilateral with 4 equal sides.

What is the formula of parallelograms? ›

The area of a parallelogram can be calculated by finding the product of its base with the altitude. The base and altitude of a parallelogram are always perpendicular to each other. The formula to calculate the area of a parallelogram is given as Area of parallelogram = base × height square units.

What is the perimeter and area of parallelogram? ›

Answer: The area A of a parallelogram is given by the formula. A=bh. Over here, b is the length of one base and h refers to the height. Further, the perimeter of a parallelogram is the sum of the lengths of its four sides.

### Why is the area of a parallelogram b times height? ›

Why is the Area of a Parallelogram Base Times Height? The area of a parallelogram is base times height because it can be rearranged into a rectangle with the same area. A rectangle is a special type of parallelogram.

How to find the area of a parallelogram given 2 sides and an angle? ›

The area A of a parallelogram, given two adjacent sides a, b and the angle C between them, is twice the area of the corresponding triangle. So A = ab sin(C).

How to find the area of a parallelogram without the height calculator? ›

The area of a parallelogram can be determined from its diagonals, provided that you also know the angle between the diagonals. If e and f are the lengths of the diagonals and φ is the angle between them, then the area can be calculated as follows: area = ½ × e × f × sin(φ) .

## Videos

1. Learn to find the area of a parallelogram
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2. Area of a Parallelogram
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3. Area of a parallelogram from a rectangle (cut-out demo)
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4. Area of a Parallelogram Using Two Vectors & The Cross Product
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5. How to Area of a Parallelogram - VividMath.com
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6. Area Of A Parallelogram
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