# Year 10+ Plane Geometry

### Chapters

### Areas of Compound Shapes

# 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

## 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

## 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

## 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:

So, Sam is paid

### 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

Learn about Plane Geometry

Author: Subject Coach

Added on: 28th Sep 2018

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