Sphere — If the radius of a sphere is doubled, by what percentage does its surface area increase?

Introduction / Context:
This question checks scaling laws for 3D shapes. For a sphere of radius r, surface area S = 4 * π * r^2. If the radius changes by a factor k, the surface area scales by k^2.

Given Data / Assumptions:

• Original radius = r
• New radius = 2r
• Surface area S = 4πr^2

Concept / Approach:
Under r -> 2r, S -> 4π(2r)^2 = 4π * 4r^2 = 16πr^2. Compare new to old to get the factor and convert to a percent increase.

Step-by-Step Solution:
Original S = 4πr^2New S' = 4π(2r)^2 = 16πr^2Increase factor = 16πr^2 / (4πr^2) = 4 ⇒ new area is 4 times the oldPercentage increase = (4 − 1) * 100% = 300%

Verification / Alternative check:
Scaling rule: areas scale with the square of linear scale (2^2 = 4×). 4× corresponds to +300%.

Why Other Options Are Wrong:
100% corresponds to a doubling, not quadrupling; 200% means 3× total, not 4×; 50% and 25% are far too small.

Common Pitfalls:
Confusing area scaling (square of factor) with length scaling (factor itself).

Final Answer:
300%