Similar Shapes Area Volume
For similar shapes, the area scale factor is the square of the length scale factor, \(k^2\). For similar solids, the volume scale factor is the cube of the length scale factor, \(k^3\). Use the correct scale factor before multiplying or dividing areas and volumes.
Common mistake: Do not use the length scale factor directly for area or volume. If the length scale factor is \(4\), the area scale factor is \(4^2 = 16\), not \(4\).
Worked example:
The two shapes are similar. The length scale factor from the smaller shape to the larger shape is 2. Calculate the area of the larger shape.
- Area scale factor is the length scale factor squared.\[2^2 = 4\]\[10 \times 4 = 40\]
Answer:
The area of the larger shape is \(40\text{ cm}^2\).
Worked example:
The two solids are similar. Calculate the length scale factor from the smaller solid to the larger solid.
- Step 1First find the volume scale factor.\[96 \div 12 = 8\]
- Step 2The length scale factor is the cube root of the volume scale factor.\[\sqrt[3]{8} = 2\]
Answer:
The length scale factor is 2.