What is the volume formula for a sphere?

The volume formula for a sphere is derived from integral calculus and geometry. It represents the amount of space that a sphere occupies. The formula is ( \frac{4}{3} \pi r^3 ), where ( r ) is the radius of the sphere.

This formula indicates that the volume of a sphere increases dramatically as the radius grows, due to the cubic relationship of radius in the formula. The factor ( \pi ) serves as a constant that connects the sphere's geometry to its volume measurement, while the fraction ( \frac{4}{3} ) arises from the integration of circular areas in three dimensions.

In contrast, the other formulas listed pertain to different geometric shapes. The formula for the volume of a cone involves the height and is different from the sphere's formula, while the formulas for a rectangular prism (length times width times height) and a rectangle (length times width) apply to polyhedral shapes, not to a three-dimensional figure like a sphere. Thus, the correct choice clearly reflects the unique properties of a sphere in three-dimensional space.