Which formula represents the surface area of a sphere?

The surface area of a sphere is represented by the formula (A = 4\pi r^2). This formula captures the two-dimensional extent of the spherical surface based on its radius (r).

To understand why this formula is appropriate, consider that the surface area is calculated based on the radius of the sphere being squared, which reflects the geometric relationship of area. The factor of (4\pi) arises from the integration of the infinitesimally small surface elements of the sphere, encompassing the entire surface area through a relationship established in calculus.

In contrast, the other choices represent different geometric properties. For instance, one option is designed for the circumference of a circle, which reflects a linear measurement rather than an area. Another represents the volume of a sphere, which involves finding the three-dimensional capacity and is defined by a different formula. Lastly, the formula including (h) pertains to the area of a circular base of a cylinder, emphasizing that it does not apply to spheres.

Therefore, (A = 4\pi r^2) is the correct representation of the surface area of a sphere due to its derivation from the geometric principles associated with spherical shapes and surfaces.