If V1, V2 are the volumes and S1, S2 are the surface areas of two cubes, then

Concept

For geometrically similar solids, if every linear dimension changes by a factor k, every area changes by k2 and every volume changes by k3.

Therefore, surface area is proportional to the two-thirds power of volume.

Application

1. Let the side lengths of the two cubes be a1 and a2.

2. For cube i, Si = 6ai2 and Vi = ai3.

3. Taking the ratio of the surface areas gives S1/S2 = a12/a22 = (a1/a2)2.

4. The volume ratio is V1/V2 = (a1/a2)3, so a1/a2 = (V1/V2)1/3.

5. Substituting this into the area ratio gives S1/S2 = (V1/V2)2/3.

Cross-check

If a1/a2 = 2, the surface-area ratio is 4 and the volume ratio is 8; 82/3 = 4, confirming the same relation.

Hence, S1/S2 = (V1/V2)2/3.