Volume Formulas

Volume measures the space a solid occupies. Standard formulas: sphere 4/3πr³, cylinder πr²h, cone 1/3πr²h, cube s³, and rectangular prism lwh. All dimensions are in consistent units — this calculator uses inches, then converts to liters, gallons, and cubic feet.

The One-Third Relationship

A cone holds exactly one-third of its circumscribed cylinder — a fact proven by Cavalieri's principle. The same relationship holds between a pyramid and its prism. This is why cone and pyramid formulas always carry the 1/3 factor.

Real-World Uses

Volume calculations size aquariums, pools, tanks, packaging, and concrete pours. When exact shape formulas don't apply, water displacement — the rise in water level when an object is submerged — measures volume directly.