What is the formula for the volume of a sphere?

The volume of a sphere is V = (4/3) × π × r³, where r is the radius. This formula was first derived by Archimedes, who proved that a sphere's volume is exactly two-thirds of the volume of the smallest cylinder that can contain it.

Why do cone and pyramid volume formulas include 1/3?

A cone has exactly one-third the volume of a cylinder with the same base and height. Similarly, a pyramid has one-third the volume of a prism with the same base and height. This 1/3 relationship is provable through Cavalieri's principle, which compares cross-sectional areas at every height.

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Volume Calculator (3D Shapes)

Calculate the volume of any 3D shape: cube, sphere, cylinder, cone, rectangular prism, and pyramid.

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Volume Formulas for Every 3D Solid

What Is Volume?

Volume is the measure of three-dimensional space a solid occupies, expressed in cubic units (cm³, m³, in³). It is critical in engineering, fluid dynamics, packaging, and construction — anywhere you need to know "how much space does this take up?"

Formula Reference

Cube: V = a³
Rectangular Prism: V = l × w × h
Sphere: V = (4/3) × π × r³
Cylinder: V = π × r² × h
Cone: V = (1/3) × π × r² × h
Square Pyramid: V = (1/3) × b² × h

The 1/3 Factor in Cones & Pyramids

Both the cone and pyramid formulas include a factor of 1/3 compared to their cylindrical/prismatic counterparts. This is because a cone is exactly one-third the volume of a cylinder with the same base and height — a relationship provable through Cavalieri's principle.