Volume

Volume is the measure of three-dimensional space enclosed by a solid. It is always in cubed units (cm³, m³, in³).

Common formulas:

• Cube: $V = s^3$
• Rectangular prism: $V = l \cdot w \cdot h$
• Cylinder: $V = \pi r^2 h$
• Sphere: $V = \tfrac{4}{3}\pi r^3$
• Cone: $V = \tfrac{1}{3}\pi r^2 h$
• Pyramid: $V = \tfrac{1}{3} \cdot B \cdot h$ ($B$ = base area)

Note the factor of 1/3 for cones and pyramids — they are exactly one-third the volume of the cylinder/prism that contains them with the same base and height.

Calculus generalises volume to arbitrary regions through triple integrals $\iiint dV$ and to solids of revolution through the disk/shell methods. The Egyptians knew the pyramid formula thousands of years before calculus existed.