Which expression gives the surface area of a sphere of radius r?

• A. 4/3 π r^3
• B. 4 π r^2
• C. π r^2
• D. 2 π r

Answer: (B)

Surface area measures how much skin the sphere has, and it should grow with the square of the radius. A neat way to see the exact form is to consider a thin spherical shell of radius r and thickness dr. The volume of that shell is its surface area times the thickness: dV ≈ S(r) dr. If you know the sphere’s volume is V = (4/3) π r^3, then dV/dr = 4 π r^2. This derivative is the surface area S(r). So the surface area is 4 π r^2. This matches the expression for the surface area. The other expressions correspond to different quantities: (4/3) π r^3 is the volume, π r^2 is the area of a circle, and 2 π r is the circumference of a circle.