Middle Years

Lesson

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?

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 $=$= $s^2$
`s`2 - Area of a rectangle $=$= $l\times w$
`l`×`w` - Area of a trapezium $=$= $\frac{1}{2}\left(a+b\right)h$12(
`a`+`b`)`h` - Area of a kite $=$= $\frac{1}{2}xy$12
`x``y` - Area of a circle $=$= $\pi r^2$π
`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 $a$`a` and $b$`b` are parallel so their average of $\frac{1}{2}\left(a+b\right)$12(`a`+`b`) 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.

Summary

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$3. If the smaller rectangle has dimensions of $x$`x` and $y$`y`, then the larger rectangle will have dimensions of $3x$3`x` and $3y$3`y`.

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

As expected, the larger rectangle's area was $9$9 times larger than the smaller rectangle's area, exactly the square of the scale factor.

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

The two rectangles in the diagram below are similar:

Find the value of $x$

`x`.

The corresponding sides of two similar triangles are $6$6 cm and $12$12 cm.

The area of the smaller triangle is $18$18 cm^{2}.

What is the area of the larger triangle?

A triangle has side lengths of $6$6 cm, $12$12 cm, and $16$16 cm.

A second similar triangle has an area that is $9$9 times larger than the first.

Find the scale factor between the two triangles.

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

The larger triangle's perimeter is the same as the smaller triangle's perimeter.

AThe larger triangle's perimeter is $9$9 times the smaller triangle's perimeter.

BThe larger triangle's perimeter is $3$3 times the smaller triangle's perimeter.

C