Solid Volume Calculator
The volume of a cone, pyramid, prism, torus, capsule, hemisphere, frustum or ellipsoid — with the size of the cube and the sphere that would hold the same amount.
Volumes fall into two families.
How the solid volume calculator works
Volumes fall into two families. Anything with parallel ends — a prism, a cylinder, a box — is simply base area × length. Anything that tapers to a point is exactly one third of the prism that contains it, which is why a cone is ⅓πr²h and a pyramid ⅓ × base × height; Archimedes proved it, and it holds for any taper whatever the base shape.
The curved solids follow from the sphere: a hemisphere is half of ⁴⁄₃πr³, a capsule is a cylinder with a sphere's worth of ends, and an ellipsoid is a sphere stretched along three axes, ⁴⁄₃πabc.
Formula: cone ⅓πr²h; pyramid ⅓lwh; torus 2π²Rr²; hemisphere ⅔πr³; ellipsoid ⁴⁄₃πabc
Worked examples
| Inputs | Volume | Note |
|---|---|---|
| A cone of radius 5 and height 12 | 314.1593 | 314.16 |
| A pyramid 6 × 6 by 9 high | 108 | 108 |
| A torus with a ring radius 10 and tube 3 | 1,776.5288 | 1,776.5 |
| A hemisphere of radius 6 | 452.3893 | 452.39 |
FAQFrequently asked questions
Why is a cone a third of a cylinder?
Because any solid that tapers evenly to a point is exactly one third of the prism or cylinder containing it — a result Archimedes proved and had carved on his tomb.
What is a frustum?
A cone or pyramid with the top sliced off parallel to the base — the shape of a bucket, a lampshade or a plant pot.
How do I turn volume into liters?
One liter is 1,000 cubic centimeters, and one cubic meter is 1,000 liters. Work in centimeters and divide by 1,000.
What is a capsule?
A cylinder with a hemisphere on each end — the shape of a pressure vessel, a pill or an LPG tank.
Does the formula change for a slanted solid?
No. A leaning cone or pyramid has the same volume as an upright one of the same base and perpendicular height — Cavalieri's principle.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.