If each edge of a cube is doubled, then the percentage increase in its total…

Concept

For similar solids, an area measure scales as the square of the linear scale factor.

If every length is multiplied by k, the total surface area is multiplied by k2.

Application

1. Let the original edge length be a. The original total surface area is S = 6a2.

2. After doubling, the edge is 2a. The new surface area is S′ = 6(2a)2 = 24a2.

3. Therefore, S′/S = 24a2/6a2 = 4, so the increase ratio is 4 − 1 = 3.

4. Percentage increase = 3 × 100% = 300%.

Cross-check

Each square face changes from area a2 to (2a)2 = 4a2. The cube still has six faces, so its total surface area also becomes four times the original; the increase is three times the original, or 300%.