Similar shapes
Similar shapes
- Two shapes are similar if one is an enlargement of the other: same angles, all sides multiplied by the same scale factor $k$.
- Shapes that are exactly the same size and shape are congruent.
Practice
Similar shapes have the same angles.
Similar shapes keep all angles equal; only the side lengths scale.
Area and volume of similar shapes
- Lengths scale by $k$, but:
$$\frac{\text{area } A}{\text{area } B} = k^2, \qquad \frac{\text{volume } A}{\text{volume } B} = k^3$$
- Worked example: similar solids with lengths $2:3$, smaller volume $40\ \text{cm}^3$.
- Volume ratio $= 2^3 : 3^3 = 8 : 27$, so larger $= 40 \times \tfrac{27}{8} = 135\ \text{cm}^3$.
Practice
Two similar shapes have lengths in the ratio 2:3. The area ratio is 4:b. What is b?
Area scales by k²: 2²:3² = 4:9, so b = 9.
Practice
Two similar solids have lengths 2:3; the smaller has volume 40 cm³. Find the larger volume (cm³).
Volume ratio 8:27, so 40 × 27/8 = 135 cm³.
You've got it
Key idea
- similar = same angles, sides $\times$ scale factor $k$
- area scales by $k^2$, volume scales by $k^3$
- lengths $2:3 \Rightarrow$ volumes $8:27$