Section 8.4: Surface Areas and Volumes of Similar Solids

Property

Two figures are similar if, and only if:

• Their corresponding angles are equal, and
• Their corresponding sides are proportional.

Both conditions must be true for figures to be similar.

Examples

• A square and a non-square rhombus both have proportional sides, but their angles are not equal. Therefore, they are not similar.
• A 3x5 rectangle and a 6x10 rectangle are similar. All their angles are $90^\circ$ (equal), and their corresponding sides are proportional ($\frac{6}{3} = \frac{10}{5} = 2$).
• Two trapezoids are similar. One has bases of length 4 and 10 and a height of 3. If the similar trapezoid has a height of 9, its scale factor is $\frac{9}{3}=3$. Its bases will be $4 \times 3=12$ and $10 \times 3=30$.

Explanation

For any two shapes to be similar, they must pass two tests. First, all their matching angles must be equal. Second, the ratios of all their matching sides must be the same. Failing either test means they are not similar.

Property

• The ratios of all corresponding dimensions in similar objects are equal.
• The corresponding angles in similar objects are equal.
• If we scale all the dimensions of a particular object by the same number, the new object will be similar to the old one.

Examples

• A cube with 2-inch sides is similar to a cube with 6-inch sides. The ratio of their sides is $1:3$.

• Two spheres are always similar to each other. A sphere with a 5 cm radius is similar to one with a 10 cm radius.