Volume Calculator

Why is cone volume one-third of cylinder volume?

A cone with the same base radius and height as a cylinder fills exactly one-third of it — a fact Archimedes proved using the method of exhaustion. Intuitively, think of pouring three full cones of liquid into a cylinder of identical dimensions: it fills exactly.

What is the difference between a pyramid and a cone?

A pyramid has a polygonal base (square, rectangular, triangular, etc.) and triangular faces meeting at an apex. A cone has a circular base with a smooth curved surface to the apex. Both share the ⅓ factor in their volume formula: V = ⅓ × base area × height.

How do I convert volume between units?

Volume scales as the cube of the linear unit. 1 m = 100 cm, so 1 m³ = 1,000,000 cm³. 1 foot = 12 inches, so 1 ft³ = 1,728 in³. Common liquid conversions: 1 liter = 1,000 cm³; 1 gallon (US) ≈ 3,785 cm³. Always cube the linear conversion factor.

What is an ellipsoid and where does it appear in nature?

An ellipsoid is a 3D generalization of an ellipse — a stretched or squashed sphere. The Earth is an oblate spheroid (a flattened ellipsoid, with equatorial radius slightly larger than polar radius). Eggs, watermelons, and cell nuclei are approximately ellipsoidal. The ellipsoid volume formula (4/3)πabc reduces to (4/3)πr³ when a = b = c = r (perfect sphere).