Volume Calculator
Find the volume and surface area of cubes, boxes, cylinders, spheres, hemispheres, cones, frustums, pyramids, triangular prisms, capsules, ellipsoids and tori — with liquid-capacity conversions (litres, US and imperial gallons, barrels) and the working shown.
Three volume examples
- Cube, side 3 m: V = 3 × 3 × 3 = 27 m³.
- Cylinder, radius 2 m, height 5 m: V = π × 2² × 5 = 62.83 m³.
- Room, 4 m × 3.5 m × 2.4 m: V = 4 × 3.5 × 2.4 = 33.6 m³.
Written by Suhaib Hassan. Every formula is standard solid geometry, and the worked examples are re-derived by hand. Our methodology.
It substitutes your dimensions into the closed-form volume and surface formulas, then converts cubic units to litres and gallons with exact factors.
Volume questions usually end in a tank, a pour or a shipment. This page goes straight from dimensions to capacity in the unit you need.
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Volume in other units
The same volume as capacity or cubic measure. Exact conversion factors.
| Unit | Volume | Typical use |
|---|
Drawing
A schematic of the solid using your proportions.
Step-by-step working
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Volume and surface-area reference
All twelve solids on this page.
| Solid | Volume | Total surface area |
|---|---|---|
| Cube (edge a) | a³ | 6a² |
| Box (l, w, h) | l·w·h | 2(lw + lh + wh) |
| Cylinder | πr²h | 2πr(r + h) |
| Sphere | (4/3)πr³ | 4πr² |
| Hemisphere | (2/3)πr³ | 3πr² (with the flat base) |
| Cone | (1/3)πr²h | πr(r + √(r² + h²)) |
| Frustum | (1/3)πh(R² + Rr + r²) | π[R² + r² + (R + r)s], s = √((R−r)² + h²) |
| Rectangular pyramid | (1/3)lwh | lw + l√(h² + w²/4) + w√(h² + l²/4) |
| Triangular prism (isosceles) | ½·b·t·L | b·t + (b + 2√(t² + b²/4))·L |
| Capsule | πr²L + (4/3)πr³ | 2πrL + 4πr² |
| Ellipsoid | (4/3)πabc | ≈ Thomsen (p = 1.6075) |
| Torus | 2π²Rr² | 4π²Rr |
Worked examples
Storage tank
A vertical tank is 3 m across (r = 1.5 m) and 5 m tall, filled to 95%.
95% fill = 33,576 L
Concrete pier
A round pier is 0.9 m in diameter and 12 m deep; order 5% extra.
with 5% extra: 8.016 m³
Capsule tank
A tank is a 4 m cylinder with hemispherical ends, r = 1.2 m.
two ends: (4/3)π × 1.728 = 7.238
V = 25.334 m³
Setting the derivative of the cylinder’s surface area to zero at constant volume gives h = 2r: the height equals the diameter.
What is volume?
Volume measures the space a solid occupies, in cubic units: a 1 m cube holds 1 m³, which is 1,000 litres. Capacity is the same idea applied to containers. Surface area, by contrast, is the total area of the solid’s outside.
Cavalieri’s principle
If two solids have the same height and every horizontal slice at the same level has equal area, the solids have equal volume. This is why a slanted cylinder has the same volume as a straight one, and why a cone is exactly one third of the cylinder with the same base and height.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Archimedes, On the Sphere and Cylinder (c. 225 BCE) — the volume and surface area of the sphere.
- Cavalieri, B. (1635). Geometria indivisibilibus continuorum.
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI) — the litre, gallon and barrel definitions used in the conversion table.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
Why is a cylinder’s surface area smallest when its height equals its diameter?
For a fixed volume V = πr²h, the surface area is 2πr² + 2V ÷ r. Setting its derivative with respect to r to zero gives h = 2r, so a cylinder that uses the least material for its volume is as tall as it is wide.
What is the difference between volume, capacity and displacement?
Volume is the space the solid occupies. Capacity is how much a container can hold, so it uses the internal dimensions. Displacement is the volume of fluid a submerged object pushes aside, which equals the object’s own volume.
Are US and imperial gallons the same?
No. A US liquid gallon is exactly 3.785411784 litres, while an imperial (UK) gallon is exactly 4.54609 litres, about 20% larger. The table shows both.
How do I find the volume of a partly filled horizontal cylinder?
The liquid’s cross-section is a circular segment of depth d in a circle of radius r, with area r²·arccos((r − d) ÷ r) − (r − d)√(2rd − d²). Multiply that area by the cylinder’s length. This calculator handles the full solid; use the Circle Calculator’s segment tool for the cross-section.
How is the ellipsoid’s surface area calculated?
Its volume, 4πabc ÷ 3, is exact. Its surface area has no elementary closed form, so the calculator uses Thomsen’s approximation with exponent 1.6075, which is accurate to about 1%.
Which units should I enter?
Use the same length unit for every dimension. The volume is in that unit cubed, and the conversion table shows it in litres, gallons, cubic feet and other units.
How box, cylinder and sphere volume are calculated: volume calculator guide.
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