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Stage 5.1-3

5.03 Scale and area

Lesson

Introduction

Between two similar figures, any corresponding distances between the larger and smaller figures will be in the same ratio, which can be referred to as the scale factor.

Much like how corresponding distances are in a fixed ratio between similar figures, corresponding areas are also in a fixed ratio.

Specifically, this ratio is equal to the square of the scale factor. But why?

Scale area

Since area is a two dimensional measurement, the calculation of area depends on two distances.

We can see this in the various area formulas:

  • Area of a square =s^2

  • Area of a rectangle =l \times w

  • Area of a trapezium =\dfrac{1}{2}\left(a+b\right)h

  • Area of a kite =\dfrac{1}{2}xy

  • Area of a circle =\pi r^2

Notice that every formula relies on two distances each (or a single distance twice, in the case of the square and circle). The trapezium also follows this rule, since the sides a and b are parallel so their average of \dfrac{1}{2}\left(a+b\right) can be considered a single distance.

As such, if the distances of a shape are multiplied by some scale factor, the area of that shape will be multiplied by that scale factor twice, once for each dimension in the formula.

The distances between similar figures must be in some fixed ratio equal to the scale factor. As such, when calculating the areas of these similar figures, the scale factor will be applied twice for the larger figure.

The ratio of areas between two similar figures is equal to the square of the scale factor.

For example:

Suppose that two rectangles are similar with a scale factor of 3. If the smaller rectangle has dimensions of x and y, then the larger rectangle will have dimensions of 3x and 3y.

Using the area formula of a rectangle, we find that the area of the smaller rectangle is xy while the area of the larger rectangle is 9xy.

Try this with some other shapes to see how the scale factor gets applied twice when calculating the area of similar figures.

Exploration

The applet below demonstrates the relationship between areas and scale factor of two similar figures.

Loading interactive...

The area of the larger figure is the area of the smaller figure multiplied by the scale factor squared.

Examples

Example 1

The two rectangles in the diagram below are similar:

Two rectangles with length of 12 centimetres and area of x centimetres squared, and 6 centimetres and 15 centimetres squared.

Find the value of x.

Worked Solution
Create a strategy

Multiply the area of the smaller rectangle by the scale factor squared.

Apply the idea
\displaystyle \text{Scale factor}\displaystyle =\displaystyle \dfrac{12}{6}Find the ratio of the side lengths
\displaystyle =\displaystyle 2Evaluate
\displaystyle x\displaystyle =\displaystyle 15\times 2^2Multiply the area by the scale factor squared
\displaystyle =\displaystyle 60Evaluate

Example 2

A triangle has side lengths of 6 \text{ cm}, 12 \text{ cm}, and 16 \text{ cm}. A second similar triangle has an area that is 9 times larger than the first.

a

Find the scale factor between the two triangles.

Worked Solution
Create a strategy

Square root the area scale factor.

Apply the idea
\displaystyle \text{Scale factor}\displaystyle =\displaystyle \sqrt{9}Square root the area scale factor
\displaystyle =\displaystyle 3Evaluate
b

What is the relationship between the perimeters of the two triangles?

A
The larger triangle's perimeter is the same as the smaller triangle's perimeter.
B
The larger triangle's perimeter is 9 times the smaller triangle's perimeter.
C
The larger triangle's perimeter is 3 times the smaller triangle's perimeter.
Worked Solution
Create a strategy

Use the scale factor from part (a).

Apply the idea

Since the scale factor between the two triangles is 3, then each side length will be 3 times longer in the larger triangle.

This means that the larger triangle's perimeter is 3 times the smaller triangle's perimeter, so the correct answer is Option C.

Idea summary

The ratio of areas between two similar figures is equal to the square of the scale factor.

Outcomes

MA5.1-11MG

describes and applies the properties of similar figures and scale drawings

MA5.2-13MG

applies trigonometry to solve problems, including problems involving bearings

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