Volume Calculator

Volume Formulas for Common Solids

Volume measures how much three-dimensional space a solid shape occupies, expressed in cubic units such as cubic inches or cubic centimeters. Unlike area, which covers a flat, two-dimensional region, volume accounts for length, width, and height, or their equivalents, all at once, which is why every volume formula multiplies at least three linear measurements together, or a squared measurement by a linear one, to arrive at a cubic result.

Each shape has its own formula, but most build on the same idea: base area times height. A cube, with all sides equal length s, has volume s³, because its square base, s², is simply stacked s units high. A cylinder follows the same logic with a circular base: its volume is πr²h, the circle’s area πr² multiplied by the height h. A sphere is the exception with its own compact formula, (4/3)πr³, which comes from adding up the area of circular cross-sections across the entire radius, a derivation calculus makes precise but that geometry students typically take as given.

Cones and pyramids share a different pattern: both are exactly one third the volume of the cylinder or prism that shares the same base and height. A cone’s volume is (1/3)πr²h, and a square pyramid’s is (1/3)b²h, where b is the base side length. That one third factor is not a coincidence or an approximation: it reflects how a pointed shape narrows to a single apex, so it encloses far less space than a shape with a flat top of the same footprint and height. The calculator above switches formulas and diagrams automatically as you change shapes, so you can compare how each one scales.

Comparing these formulas side by side also reveals why volume grows so quickly as a shape scales up. Doubling every linear dimension of a solid, its side, radius, or height, multiplies its volume by eight, not two, because each formula depends on three linear measurements multiplied together. That cubic relationship is why a slightly larger box, tank, or container can hold far more than its dimensions alone might suggest, and it is the same reasoning behind the entire family of formulas this calculator applies.