What is the formula for the surface area of a sphere?

The formula for the surface area of a sphere is given by the expression 4πr². In this formula, "r" represents the radius of the sphere. The factor of 4 arises from the geometric properties of a sphere, where the surface area is proportional to the square of its radius.

When calculating the surface area, the reasoning behind the formula comes from integrating the infinitesimally small areas that comprise the surface of the sphere. Each small area contributes to the total surface area, resulting in the final formula of 4πr², which has been derived from the principles of geometry.

The other options provided do not capture the correct relationship between the radius and the surface area of a sphere. For instance, πr² represents the area of a circle, not a sphere, while the other formulas involve factors or dimensions that do not correspond to the surface area of a spherical object. Thus, option 4πr² accurately describes how the surface area increases with the square of the radius, reinforcing the concept that as a sphere grows larger, its surface area expands significantly.