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 ( A = 4\pi r^2 ). This formula arises from the geometric properties of a sphere, where "A" represents the surface area and "r" is the radius of the sphere.

To derive this, consider the relationship between the radius of the sphere and the surface area. The constant ( 4\pi ) indicates that for every unit increase in radius, the surface area increases in a quadratic manner. Essentially, as the radius expands, the surface area grows proportionally to the square of that radius. This reflects the nature of three-dimensional space, where dimensions interact in specific mathematical ways.

The other formulas presented in the options represent either the surface area of different geometric shapes or the volume of the sphere, but do not apply to the surface area of a sphere itself. For example, ( 2\pi r^2 ) is the formula for the lateral surface area of a cylinder, while ( \frac{4}{3}\pi r^3 ) describes the volume of a sphere, and ( \pi r^2 ) is the area of a circle. Thus, the correct formula encapsulates the unique relationship between the radius