Volume Calculator
Select a 3D shape and enter its dimensions to calculate the volume.
Volume Formulas & Reference
All shapes at a glance
| Shape | Formula | Variables |
|---|---|---|
| Cube | V = s³ | s = side length |
| Rectangular Box | V = l × w × h | l = length, w = width, h = height |
| Sphere | V = (4/3)πr³ | r = radius |
| Cylinder | V = πr²h | r = radius, h = height |
| Cone | V = (1/3)πr²h | r = base radius, h = height |
| Rectangular Pyramid | V = (1/3) × l × w × h | l = length, w = width, h = height |
| Ellipsoid | V = (4/3)πabc | a, b, c = semi-axes |
What is volume?
Volume is the three-dimensional space enclosed by a solid object. It is measured in cubic units (cm³, m³, in³, ft³, etc.). Volume determines how much a container can hold (capacity) and is fundamental in physics, engineering, and everyday life.
Worked example — Cylinder
A cylindrical tank has radius r = 3 m and height h = 5 m.
- Write the formula: V = πr²h
- Substitute: V = π × 3² × 5 = π × 9 × 5
- Calculate: V = 45π ≈ 141.37 m³
The tank holds approximately 141,370 litres of water.
Worked example — Sphere
A ball has radius r = 4 cm.
- Write the formula: V = (4/3)πr³
- Substitute: V = (4/3) × π × 4³ = (4/3) × π × 64
- Calculate: V = 256π/3 ≈ 268.08 cm³
Worked example — Rectangular Box
A shipping box measures 30 cm × 20 cm × 15 cm.
- V = l × w × h = 30 × 20 × 15 = 9,000 cm³
Unit conversion quick reference
| From | To | Multiply by |
|---|---|---|
| cm³ | litres (L) | 0.001 |
| m³ | litres (L) | 1,000 |
| in³ | gallons (US) | 0.004329 |
| ft³ | gallons (US) | 7.4805 |
| ft³ | cubic yards (yd³) | 0.03704 |
References
- Weisstein, E.W. Volume. MathWorld — A Wolfram Web Resource. mathworld.wolfram.com
- Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning. (Chapter 6: Applications of Integration — Volumes)
- National Institute of Standards and Technology (NIST). SI Units — Volume. nist.gov
Related Calculators
Volume Formulas for Common Shapes
Volume measures the three-dimensional space a solid occupies, in cubic units (cm³, m³, ft³, gallons). Key formulas: Rectangular box = length × width × height. Cylinder = π × r² × h. Sphere = (4/3)π × r³. Cone = (1/3)π × r² × h (one-third of the enclosing cylinder). Pyramid = (1/3) × base area × height. The factor of 1/3 in cone and pyramid volumes reflects that they taper to a point rather than maintaining constant cross-section.
Volume calculations are used in aquarium capacity planning (gallons for fish stocking), concrete ordering (cubic yards for a foundation), fuel tank capacity (liters), medication dosing (cubic centimeters/milliliters), and packaging design (cubic inches for shipping). Note unit conversions: 1 gallon = 231 cubic inches = 3.785 liters, 1 cubic foot = 7.481 gallons, 1 liter = 1000 cm³. For irregular shapes, use water displacement (Archimedes' principle): submerge the object and measure the volume of water displaced.
Volume Formulas Summary
| Shape | Formula | Key Inputs |
|---|---|---|
| Cube | s³ | Side length s |
| Rectangular box | l × w × h | Length, width, height |
| Cylinder | π r² h | Radius r, height h |
| Sphere | (4/3)π r³ | Radius r |
| Cone | (1/3)π r² h | Radius r, height h |
| Pyramid | (1/3) × B × h | Base area B, height h |
