Volume Formulas

Cube: side cubed. Rectangular prism: length times width times height. Cylinder: pi times r squared times h — the circle area times height. Sphere: (4/3) times pi times r cubed. Cone: (1/3) times pi times r squared times h — exactly one-third of the cylinder that contains it.

That 1/3 relationship is remarkable: a cone holds exactly one-third the volume of a cylinder with the same base and height. Similarly, a pyramid is 1/3 of the prism it fits inside. This 1/3 factor appears across all pointed solids and can be proven through integration or physical experiment.