Ratios and similar shapes

Description

Two shapes are similar if one is an enlargement of the other — all corresponding angles are equal and all corresponding sides are in the same ratio (the linear scale factor $k$).

$$\frac{\text{length in image}}{\text{length in object}} = k$$

Area scale factor $= k^2$ and volume scale factor $= k^3$.

Example: Two similar triangles have corresponding sides of 6 cm and 9 cm. $$k = \frac{9}{6} = 1.5, \quad \text{area scale factor} = 1.5^2 = 2.25$$

So if the smaller triangle has area $12\text{ cm}^2$, the larger has area $12 \times 2.25 = 27\text{ cm}^2$.

Always identify the correct pair of corresponding sides when calculating the scale factor — match the sides that are in the same position in each shape.

Common error: using the linear scale factor for area (or area scale factor for length). Cube it for volume, square it for area, take the square root of the area ratio to get the length ratio.

Links

Areas and volumes