Inequalities with volumes

Inequalities with volumes compare the volumes of different 3D shapes or relate the volume of a shape to its other measurements (like surface area or side lengths). These inequalities help us understand which shapes can have the largest or smallest volume under certain conditions.

Important properties

• Among all solids with a given surface area, the sphere has the largest volume.

• Among all boxes (rectangular prisms) with a fixed surface area, the cube has the largest volume.

• If two solids have the same base area and height, and one always lies above the other, its volume is greater.

• Volume inequalities often use the AM-GM inequality for side lengths.