Which formula gives the area of a sphere?

Which formula gives the area of a sphere?

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

Answer: (A)

Think about the skin of a sphere—the surface area grows with the square of the radius, so the formula must involve r^2. When you sum up the tiny surface patches all around the sphere (or use the standard integral for a sphere’s surface), you get the total surface area as 4πr^2. The factor 4 comes from wrapping around in all directions on the sphere, with π arising from the circular cross-sections involved in the integration. This is the correct expression for the surface area because it matches how area scales with size and gives the right units and proportions when you change the radius. The other forms don’t describe the sphere’s surface area: πr^2 is the area of a circle, not a sphere; 2πr^2 isn’t the standard surface-area formula for a sphere; 4πr^3 resembles a volume expression, and the actual sphere volume is (4/3)πr^3.

Think about the skin of a sphere—the surface area grows with the square of the radius, so the formula must involve r^2. When you sum up the tiny surface patches all around the sphere (or use the standard integral for a sphere’s surface), you get the total surface area as 4πr^2. The factor 4 comes from wrapping around in all directions on the sphere, with π arising from the circular cross-sections involved in the integration.

This is the correct expression for the surface area because it matches how area scales with size and gives the right units and proportions when you change the radius. The other forms don’t describe the sphere’s surface area: πr^2 is the area of a circle, not a sphere; 2πr^2 isn’t the standard surface-area formula for a sphere; 4πr^3 resembles a volume expression, and the actual sphere volume is (4/3)πr^3.