Find the diameter of a sphere whose surface area is 81π cm2.

Concept

For any sphere, the surface area A and radius r satisfy A = 4πr2.

Since the diameter d = 2r, the same relation can also be written as A = πd2.

Application

1. The given surface area is A = 81π cm2, so 4πr2 = 81π.

2. Divide both sides by π to obtain 4r2 = 81.

3. Therefore r2 = 81/4, and the positive radius is r = 9/2 = 4.5 cm.

4. The diameter is d = 2r = 2 × 4.5 = 9 cm.

Cross-check

Using A = πd2 with d = 9 cm gives A = π × 92 = 81π cm2, which reproduces the given area.

Therefore, the diameter of the sphere is 9 cm.