How is the surface area of a sphere calculated?

The surface area of a sphere is calculated using the formula ( 4\pi r^2 ), where ( r ) represents the radius of the sphere. This formula derives from the geometry of the sphere and relates to how the sphere's surface can be imagined as wrapping around all points equidistant from its center.

In detail, the factor of ( 4 ) in the formula accounts for the spherical shape, indicating that a sphere's surface area is four times the area of a circle that has the same radius. The term ( \pi r^2 ) alone gives the area of a circle, while multiplying it by ( 4 ) reflects the additional dimensionality and the total extent of the surface area of the sphere itself.

The other options do not reflect the correct relationship for calculating the surface area of a sphere. For instance, one of the selections implies a linear measurement rather than an area, while another suggests a two-dimensional area without accounting for the three-dimensional aspect of the sphere. Understanding the geometry involved is crucial to correctly applying the formula to find a sphere's surface area.