Volume Calculator
Free volume calculator for 12 solids, from cubes and cylinders to cones, pyramids and frustums, with exact answers in π, surface area, litres and gallons.
All lengths use the chosen unit. Decimals such as 2.5 are accepted. The volume is in cubic units and the surface area in square units.
- Volume
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- Surface area
- –
- Capacity in litres
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Step by step
Volume in other units
| Unit | Volume |
|---|
How to use the volume calculator
- Pick the solid in Solid: Cube, Rectangular box (cuboid), Cylinder, Sphere, Hemisphere, Cone, Square pyramid, Rectangular pyramid, Triangular prism, Frustum of a cone, Capsule or Ellipsoid.
- Choose the Length unit of your measurements: millimetres, centimetres, metres, kilometres, inches, feet, yards or miles.
- Type the dimensions in the fields that appear. Each field shows the letter used in the formula, such as Radius (r) or Height (h). Decimals like 2.5 are accepted.
- The Volume, the Surface area and the Capacity in litres update as you type; Calculate does the same. Clear empties the fields and Copy copies the volume.
- Read Step by step for the formula, the formula with your values substituted, any slant height needed, and the surface area calculation. A diagram of the solid, with its dimensions labelled and hidden edges dashed, is drawn above the steps.
- Open Volume in other units to read the same volume in mm³, cm³, m³, km³, in³, ft³, yd³, millilitres, litres, US fluid ounces, US gallons and imperial gallons.
Results stay exact whenever possible. A sphere of radius 3 has a volume of 36π, shown with its decimal value ≈ 113.1. A cone of radius 1 and height 1 has a total surface of (1 + √2)π. When a value has no short exact form, only the decimal is shown.
Volume and surface area formulas
| Solid | Volume | Total surface area |
|---|---|---|
| Cube (edge a) | V = a³ | S = 6a² |
| Rectangular box (l, w, h) | V = l × w × h | S = 2(lw + lh + wh) |
| Cylinder (r, h) | V = π × r² × h | S = 2πr(r + h) |
| Sphere (r) | V = 4/3 × π × r³ | S = 4πr² |
| Hemisphere (r) | V = 2/3 × π × r³ | S = 3πr² (dome + flat base) |
| Cone (r, h) | V = 1/3 × π × r² × h | S = πr² + πrs, s = √(r² + h²) |
| Square pyramid (a, h) | V = 1/3 × a² × h | S = a² + 2as, s = √(h² + (a/2)²) |
| Rectangular pyramid (l, w, h) | V = 1/3 × l × w × h | S = lw + l·s₁ + w·s₂ |
| Triangular prism (b, h, L) | V = 1/2 × b × h × L | needs the three sides |
| Frustum of a cone (R, r, h) | V = 1/3 × π × h × (R² + R·r + r²) | S = π(R² + r² + (R + r)s) |
| Capsule (r, a) | V = π × r² × (4/3 × r + a) | S = 2πr(2r + a) |
| Ellipsoid (a, b, c) | V = 4/3 × π × a × b × c | approximation |
The height is always measured perpendicular to the base, not along a sloping side. For a cone or a pyramid, the length along the sloping face is the slant height s; the tool derives it from the height with Pythagoras’ theorem and shows it in the steps, because the surface area depends on it.
For a capsule (a cylinder with a half-sphere on each end), a is the length of the straight cylindrical part only, not the total length. The total length is a + 2r.
An ellipsoid’s surface has no elementary formula. This tool uses Knud Thomsen’s approximation, S ≈ 4π((aᵖbᵖ + aᵖcᵖ + bᵖcᵖ)/3)^(1/p) with p = 1.6075, which stays within about 1.1 % of the true value. When the three semi-axes are equal, the ellipsoid is a sphere and the exact 4πa² is used.
From volume to capacity
A volume in cubic units becomes a capacity with a few fixed equivalences: 1 L = 1 dm³ = 1,000 cm³, and 1 m³ = 1,000 L. A US gallon is exactly 3.785411784 L, while an imperial (UK) gallon is 4.54609 L, about 20 % larger, so always check which gallon a recipe or a spec means. A cubic foot holds about 28.3 L.
Because volume scales with the cube of length, doubling every dimension multiplies the volume by 8. The same effect makes unit conversions surprising: 1 m³ is 1,000,000 cm³, not 100.
Examples
- A cylindrical water tank 1.2 m across (radius 0.6 m) and 1.5 m tall: 0.54π ≈ 1.696 m³, about 1,696 L.
- A basketball of radius 12 cm: 2,304π ≈ 7,238 cm³, a little over 7 L of air.
- A 30 × 20 × 15 cm aquarium: 9,000 cm³, exactly 9 L.
- A lampshade shaped like a frustum with radii 15 and 10 cm and height 20 cm: about 9,948 cm³.
Frequently asked questions
What is the difference between volume and capacity?
Volume is the space a solid occupies, in cubic units such as cm³ or ft³. Capacity is how much a container holds, usually in litres or gallons. For a thin-walled container they are the same number in different units; use the inside dimensions for capacity.
Why does the answer contain π or a square root?
The tool keeps the exact value when one exists, such as 36π or (9 + 3√10)π, which is the form expected in maths exercises. For practical measurements, use the decimal after the ≈ sign.
Why is the surface area missing for the triangular prism?
The triangle’s base and height give its area, and therefore the volume, but not the lengths of its two other sides, which are needed for the rectangular faces. Add those faces separately if you know the sides.
How do I find the volume of an irregular object?
Split it into simple solids (boxes, cylinders, cones), compute each volume in the same unit and add them up, subtracting any hollow parts. For a small solid object, you can also measure the water it displaces.
Is the surface area of the hemisphere with or without the base?
With the base: 2πr² for the curved dome plus πr² for the flat disc, 3πr² in total. For the dome alone, keep two thirds of the result.
Is the diagram to scale?
Mostly. Very unequal dimensions are drawn with a ratio limited to 3 to 1 so the labels stay readable, and depth is drawn in oblique perspective. The calculations always use your real values.
Related tools
- Area Calculator Free area calculator for 11 shapes, from squares and circles to trapezoids, sectors and regular polygons, with the formula, worked steps and a labelled diagram.
- Circle Calculator Circle calculator that finds the radius, diameter, circumference and area from any one of them, plus arc length and sector area, exactly in terms of π.
- Triangle Calculator Free triangle calculator that solves any triangle from three values (SSS, SAS, ASA, AAS or SSA), with area, perimeter, heights, steps and a scaled diagram.
- Concrete Calculator Calculate the concrete volume in m³ or cubic yards and the number of 30 kg, 60 lb or 80 lb bags for a slab, footing, round column or steps.