# Areas of Compound Shapes

Compound shapes are made up of two or more simple shapes. The area of a flat shape is the amount of space it takes up. In other words, it is the size of a surface.

Let's start by recalling the formulas for the areas of some plane shapes, and then look at a few examples of how these may be combined to find the areas of compound shapes.

## Area Formulas for Simple Shapes

Triangle

$\text{Area} = \dfrac{1}{2} \times b \times h$

$b$ is the length of the base, and $h$ is the height of the triangle.

Square

$\text{Area} = s^2$

$s$ is the side-length of the square.

Rectangle

$\text{Area} = b \times h$

$b$ is the base (or length) of the rectangle.

$h$ is the height of the rectangle.

Parallelogram

$\text{Area} = b \times h$

$b$ is the base (or length) of the parallelogram.

$h$ is the height of the parallelogram.

Trapezium

$\text{Area} = \dfrac{1}{2}(a + b) \times h$

$a$ and $b$ are the parallel sides of the trapezium.

$h$ is the height of the trapezium.

Circle

$\text{Area} = \pi r^2$

$r$ is the radius of the circle.

Sector

$\text{Area} = \dfrac{1}{2} \times r^2 \times \theta$

Note: for this to work, the angle must be measured in radians

Here, $r$ is the radius and $\theta$ is the angle at the centre, in radians.

Ellipse

$\text{Area} = \pi ab$

$a$ and $b$ are half the lengths of the axes of the ellipse.

Note: In the above, the height is always perpendicular to the base of the shape.

## Example 1

What is the area of the region that is coloured blue in the picture?

Solution:

This compound shape is made up of a rectangle of base-length 6.8 cm and height 3.2 cm, and a triangle of base-length 6.8 cm and height 1.6 cm. Its area is

\begin{align*} \text{Area} &= \text{Area Rectangle} + \text{Area Triangle}\\ &= b_{\text{rectangle}} \times h_{\text{rectangle}} + \dfrac{1}{2} b_{\text{triangle}}\times h_{\text{triangle}}\\ &= 6.8 \times 3.2 + \dfrac{1}{2} (6.8)(1.6)\\ &= 21.76 + 5.44\\ &= 27.2 \text{ cm}^2 \end{align*}

## Example 2

What is the area of the region that is coloured green in the picture?

Solution:

This compound shape is made up of a square of side-length 7.2 m from which a circle of diameter 5.4 m has been cut out. The radius of the circle is $r = 5.4 \div 2 = 2.7 \text{ m}$. Its area is

\begin{align*} \text{Area} &= \text{Area Square} - \text{Area Circle}\\ &= (\text{side-length})^2 - \pi r^2\\ &= (7.2)^2 - \pi (2.7)^2\\ &\approx 28.94 \text{ m}^2 \end{align*}

## Example 3

What is the area of the region that is coloured yellow in the picture?

Solution:

This compound shape is made up of a rectangle of side-lengths 19 cm and 18 cm from which a semicircle of diameter 18 cm has been cut out, together with a trapezium of parallel side-lengths 36 cm and 18 cm, and height 9 cm. The radius of the semicircle is $r = 18 \div 2 = 9 \text{ cm}$. Its area is

\begin{align*} \text{Area} &= \text{Area Rectangle} - \text{Area Semicircle} + \text{Area Trapezium}\\ &= b_{\text{rectangle}}h_{\text{rectangle}} - \dfrac{1}{2}\pi r^2 + \dfrac{h_{\text{trapezium}}}{2}(a + b) \\ &= 19 \times 18 - \dfrac{1}{2}\pi (9)^2 + \dfrac{9}{2}(18 + 36)\\ &\approx 457.77 \text{ cm}^2 \end{align*}

## One Last Example

Sam has a part-time job, mowing the oval-shaped park shown below.

If he is paid $\0.02$ per square metre mown, how much does he earn each time he mows the park?

Solution: We need to find the area of the oval in square metres and multiply it by $\0.02$ to find Sam's earnings.

The oval is made up of two straight sides of length $60$ m, and two semi-circles of radius $\dfrac{1}{2}(40) = 20 \text{ m}$. So, its area is equal to the area of a circle of radius $20$ m, plus the area of a rectangle with base length $60$ m and height $40$ m:

\begin{align*} \text{Area} &= \pi r^2 + b \times h\\ &= \pi (20)^2 + (60)(40)\\ &= 400 \pi + 2,400 \text{ m}^2\\ &\approx 3656.64 \text{ m}^2. \end{align*}

So, Sam is paid

$\text{Wage} = 3656.64 \times 0.02 = \73.13$
for the job.

### Description

In these chapters you will learn about plane geometry topics such as

• Area (Irregular polygons, plane shapes etc)
• Perimeter
• Conic sections (Circle, Ellipse, Hyperbola etc)
• Polygons (Congruent, polygons, similar, triangles etc)
• Transformations and symmetry (Reflection, symmetry, transformations etc)

etc

Even though these chapters are marked for Year 10 or higher students, several topics are for students in Year 8 or higher

### Audience

Year 10 or higher, suitable for Year 8 + students as well.

### Learning Objectives

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