Find the exact volume of cubes, prisms, spheres, cylinders, cones, pyramids, tori, and ellipsoids — with formulas, step-by-step working and instant unit conversion.
Pick a shape and enter its dimensions — the volume updates live.
Volume
In litres
0.125
1 L = 1000 cm³
In m³
0.000125
cubic metres
In ft³
0.004414
cubic feet
In gallons (US)
0.033
≈ liquid US gal
| Shape | Volume formula | Notes |
|---|---|---|
| Cube | V = a³ | all sides equal length a |
| Rectangular Prism | V = l · w · h | any box-shaped solid |
| Sphere | V = (4/3)πr³ | r is the radius |
| Cylinder | V = πr²h | circular base, height h |
| Cone | V = (1/3)πr²h | 1/3 of cylinder of same base |
| Square Pyramid | V = (1/3)l²h | square base of side l |
| Torus | V = 2π²Rr² | R = major, r = minor radius |
| Ellipsoid | V = (4/3)πabc | a, b, c = semi-axes |
The Method
Every solid's volume can be derived by integration: slicing the shape into infinitely thin layers and summing their areas. For familiar shapes the integral collapses to a clean closed-form expression — πr²h for a cylinder, (4/3)πr³ for a sphere, and (1/3) × base × height for every cone or pyramid (the one-third factor is a consequence of integrating a quadratic profile). This calculator evaluates those closed-form formulas directly, so the result is exact within floating-point precision.
Working for selected shape
About This Tool
A volume calculator finds the amount of three-dimensional space enclosed by a solid shape. Pick one of eight common shapes — cube, rectangular prism, sphere, cylinder, cone, square pyramid, torus, or ellipsoid — enter its dimensions, and the calculator returns the exact volume in the cubic unit you choose, along with quick conversions to litres, m³, ft³ and US gallons.
Volume formulas have practical and theoretical reach. In engineering they size tanks, pipes, structural members and the dosing of materials like concrete and paint. In physics they appear in density (ρ = m/V), pressure, and fluid mechanics. In school mathematics they connect 1D length, 2D area and 3D volume through clean scaling laws — double a sphere's radius and its volume increases by a factor of 8.
The calculator uses standard closed-form formulas: V = a³ for a cube, V = πr²h for a cylinder, V = (4/3)πr³ for a sphere, V = (1/3)πr²h for a cone (one-third the matching cylinder), V = (1/3)l²h for a square pyramid, V = 2π²Rr² for a torus, and V = (4/3)πabc for an ellipsoid. Every formula is evaluated with double-precision arithmetic, accurate to about 15 significant digits.
Use this free volume calculator for homework, DIY projects, container sizing, science labs, or anywhere you need an exact volume. All computation runs in your browser — no sign-up, no tracking, no data sent to any server.
Eight Common Shapes
Cube, prism, sphere, cylinder, cone, pyramid, torus and ellipsoid in one tool.
Six Length Units
Metres, centimetres, millimetres, feet, inches, and yards — converted on the fly.
Step-by-Step Working
Each formula substitution shown — perfect for homework or revision.
Capacity Conversions
Instant litres, m³, ft³ and US gallon equivalents alongside the result.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Live, Instant Updates
Results recompute the moment you change a dimension or unit.
From shape pick to capacity conversion in under a minute.
Use the chips at the top of the card to choose from cube, box, sphere, cylinder, cone, pyramid, torus, or ellipsoid. The required input fields adapt to match.
Choose metres, centimetres, millimetres, feet, inches, or yards. The same unit applies to every dimension and the result is in the corresponding cubic unit.
Fill in each required length, radius, or semi-axis. Inputs accept decimals — for instance 3.5 — and update the volume in real time.
The headline volume shows in the cubic unit matching your length choice. Stat tiles convert it to litres, m³, ft³ and US gallons.
The bar chart shows the same volume expressed in different cubic units at a glance — useful for sanity-checking that 1 m³ really is one thousand litres.
Below the calculator, the formula working shows the exact substitution — useful for homework and verifying your hand calculation step by step.
Everything you need to know about volume, formulas, and unit conversions.
Volume is the amount of three-dimensional space enclosed by a closed surface. It is measured in cubic units — cubic metres (m³), cubic centimetres (cm³), cubic feet (ft³), and so on. For any solid shape, volume answers the question: how much could fit inside? For a hollow container the equivalent quantity is called capacity.
A sphere of radius r has volume V = (4/3)πr³. For example, a sphere with radius 5 cm has volume (4/3) × π × 125 ≈ 523.60 cm³. The formula was first derived by Archimedes around 250 BC and is one of the earliest known applications of integration.
A cylinder's volume equals its base area times its height: V = πr²h. For example a cylinder with radius 3 cm and height 10 cm has volume π × 9 × 10 ≈ 282.74 cm³. This is the same rule that gives the volume of any prism — base area × height.
A cone with the same base and height as a cylinder fits inside it exactly three times — proven both by integration and by Cavalieri's principle. So V_cone = (1/3) × π × r² × h = (1/3) × V_cylinder. The same one-third factor appears for any pyramid versus the equivalent prism, because the underlying geometry is identical.
Use cubic units consistent with your input. If lengths are in metres, volume is in m³; centimetres gives cm³; and so on. As a guide: cm³ or in³ for everyday objects, m³ or ft³ for rooms and buildings, mm³ for tiny components. The calculator handles all six common units and shows the equivalent capacity automatically.
Volume is the space occupied by a solid; capacity is the amount of liquid a container can hold. For a hollow container with negligible wall thickness the two are equal. The most useful conversion is 1 litre = 1000 cm³ = 0.001 m³. So a 2-litre bottle has an internal volume of 2000 cm³.
Volume scales with the cube of linear size. If you double every length, area increases by 4× and volume by 8×. Triple the size and volume grows 27×. This cubic scaling law is fundamental in physics and biology — it explains why elephants need thicker legs than ants and why a hailstone's mass quickly outpaces its surface drag as it grows.
A torus (doughnut) with major radius R (centre of tube to centre of torus) and minor radius r (tube radius) has volume V = 2π²Rr². This is exactly the area of the circular cross-section (πr²) times the circumference of the centre circle (2πR) — a special case of Pappus's centroid theorem.
An ellipsoid with three semi-axes a, b, c has volume V = (4/3)πabc. When a = b = c = r this reduces to the sphere formula (4/3)πr³. Ellipsoids model planets (oblate spheroids) and atomic orbitals (more elongated forms).
All formulas are evaluated in IEEE 754 double-precision floating-point, which is accurate to about 15-17 significant digits. Far more precision than any real-world measurement, so the displayed answer is limited by your input accuracy, not the calculator.