Equal surface areas — sphere vs cube, find volume ratio: A sphere and a cube have the same surface area. What is the ratio of the volume of the sphere to the volume of the cube?

Introduction / Context:
This is a comparison of two solids under an equal-surface-area constraint. Express both volumes in terms of a single parameter using the surface-area equality, then take the ratio.

Given Data / Assumptions:

• Sphere: surface area S_s = 4πr^2, volume V_s = (4/3)πr^3
• Cube: surface area S_c = 6a^2, volume V_c = a^3
• Set 4πr^2 = 6a^2 ⇒ r^2/a^2 = 3/(2π)

Concept / Approach:
Write V_s/V_c = [(4/3)π r^3]/a^3 = (4/3)π (r^2/a^2)^{3/2} and substitute r^2/a^2 from the surface-area equality.

Step-by-Step Solution:
V_s/V_c = (4/3)π * (3/(2π))^{3/2}= (4/3)π * [3√3/(2√2 * π√π)]= √6/√π

Verification / Alternative check:
Numerically, √6/√π ≈ 2.449/1.772 ≈ 1.382, a reasonable constant ratio.

Why Other Options Are Wrong:
1:2 and 3:1 are arbitrary; 6:π confuses square roots and direct factors.

Common Pitfalls:
Forgetting the 3/2 power when converting from area to volume ratios.

Final Answer:
√6 : √π