Volume Calculator
Calculate the volume and surface area of boxes, cylinders, spheres, cones, pyramids, capsules, spherical caps, conical frustums, ellipsoids and tubes, also shown in litres and gallons.
How to use this calculator
- 1Choose the solid — the form shows only the dimensions it needs.
- 2Enter all dimensions in the same unit.
- 3Read the litres figure if you are sizing a tank or container.
How the calculation works
Box: V = lwh Cylinder: V = πr²h Sphere: V = (4/3)πr³ Cone: V = ⅓πr²h Pyramid: V = ⅓b²h Frustum: V = ⅓πh(R² + Rr + r²) Ellipsoid: V = (4/3)πabc Tube: V = π(R² − r²)l- l, w, h
- Length, width and height
- r
- Radius (or the smaller radius, for a frustum or tube)
- R
- The larger radius, for a frustum or tube — or the ball radius, for a spherical cap
- b
- Base side length
- a, b, c
- The three semi-axis lengths of an ellipsoid
Cones and pyramids hold exactly one third of the prism or cylinder with the same base and height — a result that goes back to Archimedes.
Surface area for a cone uses the slant height √(r² + h²), not the vertical height; a conical frustum uses the same idea with the slant of its sloped side, √((R − r)² + h²).
A spherical cap is defined by the radius of the full sphere it was cut from (R) and the cap's own height (h) — its base radius, r = √(h(2R − h)), is derived rather than entered.
A tube (hollow cylinder) is this calculator's one shape with a cavity — volume is the outer cylinder minus the inner one, so wall thickness is already built into the result rather than assumed away.
Worked example
A cylinder with radius 0.5 m and height 2 m
- 1.Base area = π × 0.5² = 0.7854 m².
- 2.Volume = 0.7854 × 2 = 1.5708 m³.
- 3.That is 1,570.8 litres, or about 415 US gallons.
Result: 1.5708 m³ (1,570.8 litres)
A steel tube, 10 cm outer diameter, 6 cm inner diameter, 20 cm long
- 1.Outer radius 5 cm, inner radius 3 cm.
- 2.Cross-sectional area = π × (5² − 3²) = π × 16 = 50.27 cm².
- 3.Volume = 50.27 × 20 = 1,005.3 cm³.
Result: 1,005.3 cm³ (1.0053 litres)
What volume actually measures
Volume is the amount of three-dimensional space a solid takes up — how much room is inside its surface. Because it spans three dimensions, it is always expressed in cubed units (cubic metres, cubic feet, cubic centimetres), and it scales far faster than length does: double every dimension of a solid and its volume does not double, or even quadruple, but multiplies by eight.
Surface area, by contrast, is the total area of every face wrapping around that solid — a two-dimensional quantity, in squared units, that only quadruples when every dimension doubles. The two numbers answer different questions: volume asks how much a container holds, surface area asks how much material it takes to wrap or coat it.
The families of solids this calculator covers
Eleven shapes sounds like a lot, but most of them are variations on a small number of underlying ideas.
- Prisms and cuboids — a box or cube is the three-dimensional equivalent of a rectangle — volume is simply the base area multiplied straight through by the height, since every cross-section is identical.
- Cylinders and tubes — a circle extruded along a straight axis. A tube is the same idea with a smaller cylinder removed from the middle, which is why its volume is the difference between an outer and inner cylinder rather than a new formula.
- Spheres and spherical caps — a sphere is the three-dimensional analogue of a circle — every point on its surface sits the same distance from the centre. A spherical cap is simply the slice left over when a plane cuts through that sphere.
- Cones and pyramids — both come to a single point rather than running straight through, which is exactly why each holds precisely one third of the prism or cylinder sharing its base and height.
- Composite shapes — a capsule (a cylinder capped with two hemispheres) and a conical frustum (a cone with its tip sliced off) are built by combining or adjusting the simpler solids above.
From volume to capacity
For anything meant to hold a liquid or gas — a tank, a bottle, a swimming pool — volume is usually more useful expressed as capacity. The two are the same physical quantity in different units: one cubic metre holds exactly 1,000 litres, which is why converting between them is just a matter of moving the decimal point once dimensions are in metric units.
Litres and gallons are not the same size everywhere, either. A US gallon and an imperial (UK) gallon differ by about 20%, which is why a tank’s capacity can be quoted correctly as two different numbers depending on which gallon is meant.
Where volume calculations show up
Volume is the quantity behind almost every "how much fits" or "how much material" question involving a physical object.
- Shipping and packaging — freight is often priced by volume rather than weight, since a light but bulky parcel can take up as much space in a container as a heavy, dense one.
- Tanks and reservoirs — sizing a water tank, fuel tank or septic tank starts from the volume needed to hold a given capacity, usually from internal dimensions rather than external ones.
- Concrete and materials estimating — pouring a slab, footing or column requires knowing its volume to order the right amount of concrete, gravel or fill.
- HVAC and ventilation — the volume of a room sets the airflow needed to heat, cool or refresh it a given number of times per hour.
- Cooking and food production — the volume of a baking tin or mould determines batter or filling quantities, and is one of the most common places people apply volume formulas without thinking of them as geometry.
Estimating volume for irregular objects
Real containers and objects rarely match a single textbook solid exactly, so a few practical techniques cover the gap.
- 1Decompose complex shapes — break an irregular tank or object into boxes, cylinders and cones that approximate its parts, then add their volumes together — the same logic as splitting an irregular area into rectangles.
- 2Use internal dimensions for capacity — a tank’s wall thickness means its internal volume is always smaller than its external one — measure the inside if what matters is how much it can hold.
- 3Use water displacement for awkward solids — submerging an irregular object in a container of water and measuring how much the level rises gives its volume directly, without needing a formula at all.
- 4Keep every dimension in the same unit — because volume scales with the cube of length, a unit mistake here is out by a factor of a thousand rather than ten — worth double-checking before ordering materials from the result.
What this assumes, and where it stops
Assumptions
- Solids are regular. Every shape except the tube is treated as solid, with no cavity or wall thickness to deduct.
Limitations
- For tanks, this gives the internal volume only if you enter internal dimensions. Wall thickness matters for anything but thin-walled containers — the tube shape is the exception, since it takes both an inner and outer diameter.
- Irregular solids need to be decomposed into these primitives.
- Ellipsoid surface area is a well-established approximation (Knud Thomsen), not an exact formula — none exists in elementary closed form. Its volume is exact.
Common questions
How do I work out the capacity of a cylindrical tank?
Calculate the volume from the internal radius and height, then convert: one cubic metre is 1,000 litres. This calculator shows litres and gallons automatically. Remember to use internal dimensions, not external.
Why is a cone exactly one third of a cylinder?
It falls out of integrating the cross-sectional area along the height: the radius shrinks linearly, so the area shrinks quadratically, and the integral of x² from 0 to 1 is one third. Archimedes proved it geometrically over two thousand years ago and considered it his finest result.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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