Scale Factors: Length, Area, and Volume

Core Concept

Similar figures are identical in shape but different in size. Every corresponding length is multiplied by the same value — the scale factor $k$.

The cascade of consequences is the heart of this topic:

Why? Area formulas always multiply two lengths together (e.g., $A = \pi r^2$ or $A = \frac{1}{2}bh$), so each gets multiplied by $k$, giving $k \cdot k = k^2$. Volume formulas multiply three lengths (e.g., $V = \pi r^2 h$), giving $k^3$.

Quick illustration: A square has side 3, so area = 9 and perimeter = 12. Scale by $k = 4$: new side = 12, new perimeter = 48 = $12 \times 4$, new area = 144 = $9 \times 16 = 9 \times 4^2$. ✓

Finding $k$ from given information:

• If two lengths are given, $k = \dfrac{\text{new length}}{\text{original length}}$.
• If two areas are given, $k = \sqrt{\dfrac{\text{new area}}{\text{original area}}}$.
• If two volumes are given, $k = \sqrt[3]{\dfrac{\text{new volume}}{\text{original volume}}}$.