steps in solving word problems involving area of composite figures

One to one maths interventions built for KS4 success

Weekly online one to one GCSE maths revision lessons now available

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

This topic is relevant for:

Area Of Compound Shapes

Here we will learn about the area of compound shapes , including questions involving rectilinear shapes, composite shapes, missing side lengths and unit conversion.

There are also area of compound shape 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 compound shapes?

The area of compound shapes (also known as the area of composite shapes) is the amount of space inside a shape composed of basic shapes put together. It is measured in units squared ( cm^2 , m^2, mm^2 etc.).

Compound shapes can also be called composite shapes.

To find the area of compound shapes we must divide the compound shape into basic shapes and find the area of each of the basic shapes and add them together. 

Our final answer must be in square units e.g. square centimetres (cm^2) , square meters (m^2) , square feet (ft^2), etc.

How to find the area of compound shapes

In order to find the area of compound shapes:

• Break down the compound shape into basic shapes.
• Find the area of the basic shapes.
• Add the areas.

Explain how to find the area of compound shapes

Area of compound shapes worksheet

Get your free area of compound shapes worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on area

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

Area of compound shapes examples

Example 1: all side lengths given (area of the l shape).

Find the area of the compound shape below:

Break down the compound shape into basic shapes .

Split the compound shape into two rectangles. We can do this in two ways:

For the purposes of this question we will use the first way.

2 Find the area of the basic shapes .

Let us label the two rectangles A and B :

Area of Rectangle A :

\begin{aligned} \text { Area }_{A} &=\text { length } \times \text { width } \\\\ \text { Area }_{A} &=6 \times 5 \\\\ &=30 \end{aligned}

Area of Rectangle B :

\begin{aligned} \text { Area }_{B} &=\text { length } \times \text { width } \\\\ \text { Area }_{B} &=9 \times 4 \\\\ &=36 \end{aligned}

3 Add the areas .

\begin{aligned} \text { Compound Area } &=\text { Area of } A+\text { Area of } B \\\\ &= 30+36 \\\\ &=66 \end{aligned}

4 Write down your final answer with units squared .

\text { Area }=66 \mathrm{~m}^{2}

Example 2: missing lengths 

Split the compound shape into two rectangles and find the area of each rectangle:

Find the area of the basic shapes .

Let us label the two rectangles A and B .

We need to calculate the unknown length. To do this we need to use the measurements we are given in the question that are parallel to the unknown side:

If we add together the 3 sides, we will get the length of rectangle A .

\begin{aligned} \text { Area }_{A} &=\text { length } \times \text { width } \\\\ \text { Area }_{A} &=14 \times 2 \\\\ &=28 \end{aligned}

The length of the entire compound shape is 10m . We need to subtract the 2m of shape A to get just the length of shape B .

\begin{aligned} \text { Area }_{B} &=\text { length } \times \text { width } \\\\ \text { Area }_{B} &=8 \times 5 \\\\ &=40 \end{aligned}

Add the areas .

\begin{aligned} \text { Compound Area } &=\text { Area of } A+\text { Area of } B \\\\ &=28+40 \\\\ &=68 \end{aligned}

Write down your final answer with units squared .

\text { Area }=68 \mathrm{~m}^{2}

Example 3: worded question

The wall shown below needs to be painted. Each tin of paint covers a distance of 4 square metres. How many tins of paint will be needed to cover the entire wall?

Split the compound shape into three rectangles:

Let us label the two rectangles A, B and C .

\begin{aligned} \text { Area }_{A} &=\text { length } \times \text { width } \\\\ \text { Area }_{A} &=12 \times 5 \\\\ &=60 \end{aligned}

Rectangle C is congruent to Rectangle A so we can just double the area of rectangle A .

60 \times 2 = 120

We need to calculate the unknown width. To do this we need to use the measurements we are given in the question that are parallel to the unknown side:

The entire width of the shape is 13m . If we subtract 5m and 5m (the width of rectangles A and C ).

\begin{aligned} \text { Area }_{B} &=\text { length } \times \text { width } \\\\ \text { Area }_{B} &=7 \times 3 \\\\ &=21 \end{aligned}

\begin{aligned} \text { Compound Area} &=\text { Area of } A+\text { Area of } B+\text { Area of } C \\\\ &=60+21+60 \\\\ &=141 \end{aligned}

\text { Area }=141 \mathrm{~m}^{2}

While this is the total area we need to calculate how many tins of paint will be needed to cover the wall. If each tin of paint covers a distance of 4 square metres, we need to divide 4 into 141 .

141 \div 4=35.25

Since you cannot buy 0.25 of a can of paint, we need to round our answer to 36 tins of paint.

Answer = 36 tins of paint will be needed

Example 4: compound shapes involving non-rectangular shapes

Split the compound shape into a triangle and rectangle:

*Note: This shape is a trapezium. Another way to find the area of this shape would be simply to use the formula for the area of a trapezium. For this question, because we are focusing on compound shapes, we will separate the shape into a triangle and a rectangle.

Let us label the two shapes A and B .

Area of Triangle A :

\begin{aligned} \text { Area }_{A} &=\frac{b \times h}{2} \\\\ \text { Area }_{A} &=\frac{2 \times 8}{2} \\\\ &=8 \end{aligned}

The height of the entire compound shape is 6m . We need to subtract the 2m of shape A to get just the length of shape B .

\begin{aligned} \text { Area }_{B} &=\text { length } \times \text { width } \\\\ \text { Area }_{B} &=8 \times 4 \\\\ &=32 \end{aligned}

\begin{aligned} \text { Compound Area } &=\text { Area of } A+\text { Area of } B \\\\ &=8+32 \\\\ &=40 \end{aligned}

\text { Area }=40 \mathrm{~m}^{2}

Example 5: missing side lengths

Calculate the value of x in the compound shape below:

Split the compound shape into two rectangles. 

Area of Rectangle A:

\begin{aligned} \text { Area }_{A} &=\text { length } \times \text { width } \\\\ \text { Area }_{A} &=8 \times 2 \\\\ &=16 \end{aligned}

In this case we are not given enough information to find the area of Rectangle B. .

\begin{aligned} \text { Compound Area } &=\text { Area of } A+\text { Area of } B \\\\ 48 &=16+\text { Area of } B \\\\ \text { Area of } B &=48-16 \\\\ \text { Area of } B &=32 \end{aligned}

Now that we know the area of rectangle B, we can calculate the value of x .

\begin{aligned} \text { Area of } B &=l \times w \\\\ 32 &=l \times 4 \\\\ l &=\frac{32}{4} \\\\ l &=8 m \end{aligned}

Example 6: worded question

Below is a blueprint for a garden to be created in the middle of town. It will feature a triangular flowerbed and the remainder of the garden will be covered in grass.

Each roll of grass costs £24 and covers a distance of 3 square metres. How much will it cost to cover the garden with grass?

Split the compound shape into a rectangle and trapezium. Once we find the area of the two shapes we can subtract the area of the triangle. 

Let us label the three shapes A , B and C .

Area of Trapezium A :

Add the parallel sides: 10+11=21

Divide by 2: 21 \div 2=10.5

Times by the height: 10.5 \times 2=21m^2

Step by step guide: Area of a trapezium (Coming soon)

\begin{aligned} \text { Area }_{B} &=\text { length } \times \text { width } \\\\ \text { Area }_{B} &=11 \times 7 \\\\ &=77 \mathrm{~m}^{2} \end{aligned}

Area of Triangle C :

The width of the entire compound shape is 11m . We need to subtract the 5m and 4m from 11m to get the width of shape C .

\begin{aligned} \text { Area }_{C} &=\frac{1}{2}(\text { base } \times \text { height }) \\\\ \text { Area }_{C} &=\frac{2 \times 6}{2} \\\\ &=6 m^{2} \end{aligned}

\begin{aligned} \text { Compound Area } &=\text { Area of } A+\text { Area of } B-\text { Area of } C \\\\ &=21+77-6 \\\\ &=92 \end{aligned}

\text { Area }=92 \mathrm{~m}^{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 rectangle, pupils must look at the units given in the question. If different units are given e.g. length = 4m and width = 3cm pupils must convert them either both to cm or both to m.

• Composite shapes

A composite shapes is the same as a compound shape

Practice area of compound shapes questions

1. Find the area of the compound shape below:

Split into two rectangles:

Area of A : 10 \times 5=50

Area of B : 4 \times 4=16

Compound Area: 50+16=66m^2

2. Find the area of the compound shape below:

Area of  A : 10 \times 5=50

Area of  B : 9 \times 6=54

Compound Area: 50+16=104m^2

3. The wall shown below needs to be painted. Each tin of paint covers a distance of 4 square metres. How many tins of paint will be needed to cover the entire wall?

Split into three rectangles:

Area of A : 15 \times 5=75

Area of B : 8 \times 2=16

Area of C : 15 \times 5=75

Compound Area: 75+16+75=166m^2

Then divide by 4 to get the number of tins.

4. Find the area of the compound shape below:

Split into one rectangle and two triangles:    

Area of A : \frac{1}{2}(190\times80)=7600

Area of B : \frac{1}{2}(190\times80)=7600

Area of C : 170 \times 380=64600

Compound Area: 7600+7600+646000=79,800m^2

5. Calculate the value of x in the compound shape below:

6. An architect is developing a blueprint for one floor of a house which is shown below. The shaded region represents the floor and will need to be fully tiled. Each square tile has a width of 12cm . How many tiles will be needed to cover the entire floor?

Area of A : 7 \times 11=77

Area of B : \frac{1}{2}(4\times3)=6   \begin{aligned} &77-6=71m^2 \\ &12cm = 0.12m \end{aligned}     Area of 1 tile: = 0.12 \times 0.12=0.0144m^2   \begin{aligned} &71\div0.0144=4930.56\\ \end{aligned}   4931 tiles

Area of compound shapes GCSE questions

1. The diagram below shows a shape. Work out the area of the shape.

6 \times 7=42 seen

15\times5=75 seen or 5m seen as the missing vertical length

117m^2 (cao including units)

2. (a) A farmer owns the cattle field shown below. Each cattle costs approximately £450 and the farmer estimates that each cow requires a minimum of 4 square metres to graze freely.

What is the maximum number of cattle that could fit on the field?

(b)      How much would it cost to acquire all the cattle?

12 \times 5=60 seen (area of the rectangle)

  \frac{1}{2}(12\times3)=18 seen (area of the triangle)

  60+18=78 seen (total area)

  78\div4=19.5

  19 (cao)

  19\times450

  £8550

3. What is the perimeter of the shape below?

Splits into two rectangles and calculates \times by using vertical side lengths given (20-7) or 13 seen

Calculates area of A by multiplying 13 \times 8 or 104 seen

132-104 = 28 seen B = 28cm^2

  z=28\div7, \; z = 4

  Using horizontal lengths to calculate y, \; y=8-4, \; y=4

P=8+20+4+13+4+7 or 56cm seen

Learning checklist

You have now learned how to:

• Calculate and solve problems involving composite shapes
• Calculate and solve problems involving the perimeter and area of triangles, parallelograms and trapezia

The next lessons are

• How to work out perimeter
• 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 tuition programme.

Privacy Overview

Area of Composite Figures Word Problems

The area of composite shapes is the area that is covered by any composite shape. The composite shape is a shape in which few polygons are put together to form a required shape. These shapes or figures can be made up of a combination of triangles, squares, and quadrilaterals, etc. Divide a composite shape into basic shapes like square, triangle, rectangle, hexagon, etc. to determine the area of composite shapes.

1. Read the word problem and decompose the composite figure into simpler shapes of which you can find the area. 2. Write down the area formulas of the simpler shapes. 3. Substitute the given values into the formulas. 4. Add or subtract the areas of the simple shapes as needed and state the correct square units.

The area is composed of a square and two semicircles (one whole circle).

Area of the pool = Area of a circle + Area of a square

= πr 2 + s 2

= 3.14 (18.25 m) 2 + (36.5 m) 2

= 1045.81 m 2 + 1332.25 m 2

= 2378.1 m 2

Practice Area of Composite Figures Word Problems

Practice Problem 1

Practice Problem 2

Practice Problem 3

Composite figure – a figure created by adding two or more shapes.

Area – the space occupied by a flat shape or the surface of an object.

Pre-requisite Skills Measurement – Perimeter Measurement – Area Circumference and Area of Circles Circles: Diameter, Radius, Circumference Area of a parallelogram Area of a triangle

Related Skills Surface Area of Prisms Surface Area of Pyramids Surface Area of Cylinders Surface Area of Cones Surface Area of Rectangular Prisms Surface Area of Composite Figures

Determining Area of Composite Figures

Introduction.

As you have seen in previous grades or lessons, the area of a figure is the amount of flat space that the figure covers. Area is measured in square units, such as square inches or square centimeters.

Area formulas with which you may be familiar include the following.

Not all figures that you will encounter can be neatly described as one of these polygons or as a circle. Instead, they may be part of a figure, such as a semicircle , or they may be made up of several polygons. If a figure is made up of several polygons, parts of polygons, or parts of circles, then that figure is called a composite figure . In this lesson, you will learn how to determine the area of composite figures.

Breaking Down Composite Figures

In the introduction, you reviewed area formulas that you can use to calculate the area of particular polygons and circles. In the real world, not every figure can be cleanly classified as a rectangle, parallelogram, triangle, or trapezoid. Consider the following real-world objects for which it is useful to be able to determine the area.

Source: Landscape Design Advisor, Flickr

Source: Liben(121), Dezidor, Wikimedia Commons

In order to calculate the area of these figures and many more, you need to be able to solve problems that involve composite figures .

The border of a playground is shown below. Look at the second slide to see how one student partitioned the figure.

One way that you can break down a figure into triangles, squares, and parallelograms is to use tangrams. Tangrams are tiles from a Chinese puzzle that can be rearranged to create geometric designs.

Need additional directions?

Pause and Reflect

• When you are partitioning a composite figure into triangles, what patterns should you look for?
• If you were partitioning a figure into rectangles, what would you need to look for?

For questions 1–3, partition the composite figure into a combination of rectangles, triangles, parallelograms, trapezoids, or circles (including semicircles or quarter circles). There may be more than one possible answer. You may assume that angles appearing to be right angles are right angles.

Matching Area Formulas with Figures

In the last section, you practiced different ways to partition a composite figure into rectangles , triangles , parallelograms , trapezoids , squares , circles , semicircles , or quarter circles . In this section, you will take those smaller parts and identify area formulas necessary to calculate the area of each component region of the composite figure.

A swimming pool company offers several designs of swimming pools, including the ones shown below.

Nedra is in charge of calculating the area of the surface of each pool. The animation below shows how Nedra divided the composite figure representing each swimming pool into regions. These regions should have simple area formulas. See how she did it below.

The actual dimensions for each pool are shown below. Calculate the area of each pool by calculating the area of each component region, and then determine the sum of the areas.

Check Your Answer Pool #1 Check Your Answer Pool #2 Check Your Answer Pool #3

• Thinking about area formulas, which type of polygon or circle is it easier to break a composite figure into in order to calculate the area?
• Why is it sometimes possible to break a composite figure into different polygons?

For questions 1–3, each composite figure is broken into different component regions. Identify the area formula required to calculate the area of each component region.

Solving Problems Involving Composite Figures

In the last two sections, you practiced breaking a composite figure into component regions. These regions should have simple area formulas. In this section, you will apply that knowledge to solving both mathematical and real-world problems.

Sometimes, a composite figure contains a hole or a region that must be removed when determining the area of the composite figure.

What operation describes removing, or taking away, a quantity?

Use the interactive below to practice solving problems involving composite figures that are made up of polygons or parts of circles. On screen directions will be provided. You will be asked to identify the component polygons and/or circles. Identify the necessary area formulas, and calculate the area of each component polygon and/or circle. Work through several problems before returning to the lesson. When you click the image below, the interactive will appear in a new browser tab or window.

Sometimes, you have to remove a region from a composite figure when you are determining its area.

Watch the video below to see how to use subtraction when determining the area of a composite figure.

How can you tell whether you need to add or subtract areas of component regions of composite figures?

In this lesson, you learned how to apply what you know about the area of triangles, certain quadrilaterals, and circles to solving problems involving composite figures . A composite figure is a shape whose classification is not readily apparent, but whose area can be determined by breaking it into component regions with simple area formulas.

The area of composite figures can be determined using either addition or subtraction.

Copy and paste the link code above.

Related Items

Area of Composite Shapes

The area of composite shapes is the area that is covered by any composite shape. The composite shape is a shape in which few polygons are put together to form a required shape. These shapes or figures can be made up of a combination of triangles, squares, and quadrilaterals, etc. Divide a composite shape into basic shapes like square, triangle, rectangle, hexagon, etc. to determine the area of composite shapes.

Basically, a composite shape is made up of basic shapes put together. It is also called a "composite" or "complex" shape. The area of the composite shape is explained in this mini-lesson along with solved examples and practice questions.

What is the Area of Composite Shapes?

The area of the composite shapes is the area of combined shapes of one or more simple polygons and circles . To calculate the area of the composite shapes we can add the areas of all the basic shapes together. In order to find the area of composite shapes, simply find the area of each shape and add them together. The following figure will give an idea about finding the area of a composite shape. The unit of the area of composite shapes is expressed in terms of m 2 , cm 2 , in 2 or ft 2 , etc.

How to Find Area of Composite Shapes?

The area of composite shapes is a combination of basic shapes . By the following steps mentioned below, we can calculate the area of the composite shapes.

• Step 1 : Break the compound shape into basic shapes.
• Step 2 : Find the area of each and every basic shape.
• Step 3 : Add all the areas of basic shapes together.
• Step 4 : Represent the answer in square units.

In order to decompose any composite shape, we must know to calculate the area of some basic shapes like squares, triangles, rectangles, and so on. Check the table below containing the area of the basic shapes.

Example: Find the area of the composite shape which is formed by joining a square and a triangle. The length of the side of the square is 5 units. The base and height of the triangle are 6 units and 7 units respectively.

Solution: Given the length of the side of the square = 5 units, the base of the triangle = 6 units, and the height of the triangle = 7 units

Area of composite shape = Area of square + Area of triangle ⇒ A = (5) 2 + [(1/2) × 6 × 7] ⇒ A = 25 + 21 = 46 square units

The area of the composite shape is 46 square units.

Solved Examples on Area of Composite Shapes

Solution : The length and breadth of rectangle ABCD are 2 in and 7 in. The length of the side of the square DEFG is 3 in.

Using the formula for the area of the composite shape,

Area of composite shape = Area of rectangle + area of the square. ⇒ Area of composite shape = Length × Breadth + side 2 ⇒ Area of composite shape = BC × AB + DE 2 ⇒ Area of composite shape = 2 × 7 + 3 2 ⇒ Area of composite shape = 14+9 = 23 square inches.

Therefore, the area of the given composite shape is 23 square inches.

Example 2: A composite shape has an area of 500 units square. The shape is composed of a circle and a triangle and the area of the triangle is 350 units square. What is the area of the circle?

Solution: Given the area of a composite shape = 500 units square and area of triangle = 350 units square

Using the formula for the area of the composite shape, Area of composite shape = area of triangle + area of the circle. ⇒ 500 = 350 + area of circle ⇒ Area of circle = 500 - 350 ⇒ Area of the circle = 150 units square.

Therefore, the area of the circle is 150 units square.

go to slide go to slide

Book a Free Trial Class

Practice Questions on Area of Composite Shapes

Faqs on area of composite shapes.

The area of composite shapes is defined as the area covered by any composite shape. A composite shape is made up of basic shapes put together. Thus, the area of the composite shape is found by individually adding all the basic shapes.

What is the Area of Composite Shapes Formula?

There is no fixed formula to determine the area of composite shapes. The area of composite shapes can be calculated by dividing the composite shape into basic shapes like square, triangle, circle, rectangle, polygon, etc. and add their areas.

What is the Unit of the Area of Composite Shapes?

The area of composite shapes can be found out by adding all the areas of figures together. The unit of the area of composite shapes is expressed in square units like m 2 , cm 2 , in 2 or ft 2 , etc.

How to Find the Area of Composite Shapes?

The steps for finding the area of composite shapes are:

• Step 1 : Divide the compound shape into basic shapes.
• Step 2 : Find the area of each basic shape separately.
• Step 4 : Now, write the answer in square units.

How to Find the Area of Composite Shapes If the Areas of All the Basic Shapes In It are Known?

The area of composite shapes if the areas of all the basic shapes in it are known is found by using the following steps:

• Step 1 : Identify the individual area of all basic shapes.
• Step 2 : Add the areas of all basic shapes together.
• Step 3 : Now, write the answer in square units.

What Happens to the Area of Composite Shapes If the Dimensions of all the Basic Shapes are Increased?

If the dimensions of all the basic shapes are increased, the area of composite shapes also increases. The area of composite shape changes as it depends on the individual area of basic shape which gets changed when the dimensions are increased.

FINDING THE AREA OF A COMPOSITE FIGURE WORKSHEET

Problem 1 :

Find the area of the figure given below.

Problem 2 :

Problem 3 :

1. Answer :

Separate the figure into smaller, familiar figures: a parallelogram and a trapezoid.

Find the area the parallelogram.

Base (b) = 10 cm

Height (h) = 1.5 cm

Use the formula.

A = 10 · 1.5

The area of the parallelogram is 15 square cm.

Find the area the trapezoid.

Base 1  (b 1 )  = 7 cm

Base 2  (b 2 )  = 10 cm

A = (1/2)h ( b 1  +  b 2 )

= (1/2)(1.5) (7  + 10 )

= (1/2)(1.5) (17 )

The area of the trapezoid is 12.75 square cm.

Add the areas to find the total area.

A = 15 + 12.75

So, the area of the given composite figure is 27.75.

2. Answer :

Separate the figure into smaller, familiar figures: a two triangles and a rectangle.

Find the area the first triangle.

Base (b) = 3 ft

Height (h) = 2 ft

A = (1/2)bh

= (1/2)(3)(2)

The area of the first triangle is 3 square ft.

Find the area the rectangle.

Length (l) = 8 + 3  =  11 ft

Height (h) = 4 ft

The area of the rectangle is 44 square ft.

Find the area the second triangle.

Height (h) = 3 ft

= (1/2)(3)(3)

The area of the second triangle is 4.5 square ft.

A = 3 + 44 + 4.5

So, the area of the given composite figure is 51.5 square feet.

3. Answer :

Separate the figure into smaller, familiar figures: a square and a semicircle.

Find the area the square.

Length of each side = 10 m

A = Side x Side

The area of the rectangle is 100 square meter.

Find the area the semicircle.

Diameter = 10 m

Radius (r) = Diameter/2

A = (1/2)πr 2

= (1/2)(3.14)(5) 2

= 1.57 x 25

The area of the semi circle is about 39.25 square meter.

A = 100 + 39.25

So, the area of the given composite figure is about 139.25 square meter.

Kindly mail your feedback to   [email protected]

We always appreciate your feedback.

© All rights reserved. onlinemath4all.com

• Sat Math Practice
• SAT Math Worksheets
• PEMDAS Rule
• BODMAS rule
• GEMDAS Order of Operations
• Math Calculators
• Transformations of Functions
• Order of rotational symmetry
• Lines of symmetry
• Compound Angles
• Quantitative Aptitude Tricks
• Trigonometric ratio table
• Word Problems
• Times Table Shortcuts
• 10th CBSE solution
• PSAT Math Preparation
• Privacy Policy
• Laws of Exponents

Recent Articles

Endpoints of Diameter of a Circle

Nov 27, 23 05:42 PM

Solving Difficult Problems Using Elimination Method

Nov 25, 23 01:52 AM

Elimination Method Examples

Nov 24, 23 07:35 AM