Which formula gives the surface area of a sphere with radius r?

• A. 2 pi r^2
• B. 4 pi r^2
• C. pi r^2
• D. 4 pi r^3

Answer: (B)

The surface area of a sphere scales with the square of the radius, and the exact factor is found by integrating the area element over the whole surface. For a sphere, each tiny patch contributes dS = r^2 sinθ dθ dφ in spherical coordinates. When you integrate θ from 0 to π and φ from 0 to 2π, the angular part gives 4π, and you’re left with S = 4πr^2. This matches the idea that surface area grows with r^2, not r^3. A quick cross-check comes from the volume formula V = (4/3)πr^3; differentiating with respect to r gives dV/dr = 4πr^2, which is the same expression as the surface area. So the correct formula is 4πr^2. The other options don’t fit: 2πr^2 would be the curved surface area of a hemisphere, πr^2 is the area of a circle, and 4πr^3 has the wrong r-power and units.