Volume formula chart
The volume formulas for the common three-dimensional solids. Volume measures the space a solid occupies and is given in cubic units. Identify your solid, then substitute its dimensions into the matching formula.
Volume formulas
| Solid | Volume formula |
|---|---|
| Cube | V = s³ |
| Rectangular prism | V = l × w × h |
| Cylinder | V = πr²h |
| Sphere | V = (4/3)πr³ |
| Cone | V = (1/3)πr²h |
| Pyramid | V = (1/3) × base area × h |
| Triangular prism | V = ½ × b × h × length |
| Ellipsoid | V = (4/3)π × a × b × c |
Volume is measured in cubic units. A useful pattern: a cone holds exactly one third of the cylinder that encloses it, and a pyramid holds one third of its enclosing prism, which is where those one-third factors come from. A sphere fills two thirds of the cylinder that just contains it.
Need areas or circle measurements?
See the Area Formula Chart and the Circle Formula Chart.
Prisms, cylinders, and the one-third rule
A prism or cylinder volume is simply its base area times its height. The pointed solids, cones and pyramids, hold exactly one third of the prism or cylinder with the same base and height. This one-third factor is not arbitrary; it falls out of the geometry and is the reason those formulas look like the flat-topped ones divided by three.
The sphere
A sphere of radius r has volume four thirds pi r cubed. A neat relationship ties it to the cylinder that just encloses it: the sphere fills exactly two thirds of that cylinder volume, a result first found by Archimedes, who was so pleased with it that he asked for a sphere and cylinder on his tomb.
FAQ
What is the volume formula for a cylinder?
Pi times the radius squared times the height, V = πr²h. It is the circular base area multiplied by how tall the cylinder is.
What is the volume of a sphere?
Four thirds pi times the radius cubed, V = (4/3)πr³. For a radius of 3, that is about 113 cubic units.
How is a cone volume related to a cylinder?
A cone holds exactly one third of the cylinder with the same base and height, which is why its formula is the cylinder formula divided by three.