Quick answer
Volume is the amount of three-dimensional space a shape occupies, measured in cubic units. Pick the shape that matches your object, enter its dimensions, and the calculator applies the matching formula — a³ for a cube, (4/3)πr³ for a sphere, πr²h for a cylinder, and so on — showing the base area and every calculation step along the way.
Key takeaways
- Volume is always expressed in cubic units — cm³, m³, in³, ft³ — never square or linear units.
- Cones and pyramids hold exactly 1/3 the volume of a cylinder or prism sharing the same base and height.
- Doubling every dimension of a shape multiplies its volume by 8 (2³), not 2.
- 1 liter equals exactly 1000 cubic centimeters, which makes metric volume-to-capacity conversion straightforward.
- Converting a linear unit conversion factor into a volume conversion factor requires cubing it, not applying it directly.
Volume formulas for every shape
| Shape | Formula | Needs |
|---|---|---|
| Cube | V = a³ | Side length a |
| Sphere | V = (4/3)πr³ | Radius r |
| Cylinder | V = πr²h | Radius r, height h |
| Cone | V = (1/3)πr²h | Radius r, height h |
| Rectangular prism | V = lwh | Length, width, height |
| Triangular prism | V = ½ × base × height × length | Triangle base, height, prism length |
| Square pyramid | V = (1/3)a²h | Base side a, height h |
| Ellipsoid | V = (4/3)πabc | Three semi-axes a, b, c |
| Torus | V = 2π²Rr² | Major radius R, minor radius r |
Worked examples
Cylinder: radius = 2, height = 5
- Base area = πr² = π × 2² = 4π ≈ 12.566
- Volume = base area × height = 4π × 5 = 20π ≈ 62.832
Cone: radius = 3, height = 4
- Base area = πr² = π × 3² = 9π ≈ 28.274
- Volume = (1/3) × base area × height = (1/3) × 9π × 4 = 12π ≈ 37.699
Notice the cone holds exactly one-third the volume of a cylinder sharing the same radius and height (12π vs. 36π) — this 1/3 relationship holds for any cone-and-cylinder or pyramid-and-prism pair with matching bases.
Cubic units and liquid capacity
Volume in metric units converts directly to liquid capacity: 1 cubic centimeter (cm³) equals 1 milliliter, and 1000 cm³ equals 1 liter. A rectangular fish tank measuring 50 cm × 30 cm × 40 cm holds:
V = 50 × 30 × 40 = 60,000 cm³ = 60 liters
In US customary units, 1 cubic foot ≈ 7.48 gallons, and 1 US gallon ≈ 231 cubic inches — handy conversions when sizing tanks, containers, or shipping volume.
Common mistakes
- Forgetting the 1/3 factor for cones and pyramids — a very easy formula to drop under time pressure.
- Entering the diameter where the formula expects the radius (radius is always half the diameter).
- Converting length units directly instead of cubing the conversion factor — 1 m = 100 cm, but 1 m³ = 1,000,000 cm³ (100³), not 100 cm³.
- Reporting volume in square units (like cm²) instead of cubic units (cm³) — a leftover habit from area calculations.
Related calculators
- Surface Area Calculator — for the outer surface area of the same 3D shapes
- Area Calculator — for two-dimensional shapes like circles, triangles, and rectangles
- Circle Calculator — for a circle's radius, diameter, circumference, and area
- Pythagorean Theorem Calculator — for right-triangle side lengths, useful when deriving prism dimensions