6.03 Scale and 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.

Examples

Example 1

The two rectangles in the diagram below are similar:

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 2	Evaluate
\displaystyle x	\displaystyle =	\displaystyle 15\times 2^2	Multiply the area by the scale factor squared
	\displaystyle =	\displaystyle 60	Evaluate

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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 3	Evaluate

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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.

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