The surface area of a sphere with radius r is which expression?

• A. 2πr^2
• B. 4πr^2
• C. πr^3
• D. πr

Answer: (B)

The surface area of a sphere grows with the square of the radius, and the exact amount comes from summing the areas of many thin belts around the sphere. Imagine slicing the sphere into tiny horizontal belts. Each belt has a circumference 2π times its radius (which at that belt is r sin θ) and a small width along the surface of r dθ. So the belt’s area is 2π (r sin θ) (r dθ) = 2π r^2 sin θ dθ. Integrating from the bottom to the top (θ from 0 to π) gives S = ∫₀^π 2π r^2 sin θ dθ = 2π r^2 [−cos θ]₀^π = 2π r^2 (1 − (−1)) = 4π r^2.

So the surface area is 4πr^2. This also matches the intuition that the sphere’s surface completely wraps around in all directions, and the result scales as r^2. The other expressions don’t fit because they either give the wrong scaling or incorrect units.