Volume of 3D Shapes Calculator: Cylinders, Cones and Frustums

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How to use this tool

Choose the solid and enter its dimensions. The result is the volume, with the surface area or slant length alongside where that is useful. Units follow your inputs: metres in, cubic metres out.

The one to know about is the frustum, a cone with the top sliced off. That is the true shape of a conical stockpile that has been partly loaded out, a battered circular excavation, and a tapered pier. People routinely approximate it as a cylinder on the average diameter, and the error from doing that is not small.

The one third rule, and why it keeps appearing

A cone is exactly one third of the cylinder that encloses it, and a pyramid is exactly one third of its box. That is not a coincidence of two formulas: any solid that tapers linearly from a base to a point takes a third of the prism on the same base. It is worth internalising because it lets you sanity check answers without a calculator: a cone of radius 2 and height 3 must come to a third of its cylinder, and it does.

The frustum interpolates between those worlds. Its volume is pi h over 3 times R squared plus R r plus r squared, and the middle term is what the average diameter shortcut throws away. Set r equal to R and the formula collapses to the cylinder; set r to zero and it collapses to the cone. Both collapses were verified exactly, which is the strongest check a frustum implementation can pass.

A worked check you can repeat on paper

A stockpile 20 m across at the base, 8 m across the flattened top, 5 m high. R is 10, r is 4, so R squared plus R r plus r squared is 100 plus 40 plus 16, which is 156. Volume is pi times 5 over 3 times 156, about 817 cubic metres.

Now see what the average diameter shortcut would have said: average radius 7, cylinder volume pi times 49 times 5, about 770 cubic metres. Six percent light, every time, always in the same direction, because the average radius squared is always less than the average of the squares. On a thousand tonne pile that is real material.

Mistakes that catch people out

Approximating a frustum by its average diameter. Systematically underestimates, as above. Fix: use the three term formula, which is what this tool does.

Entering diameters where the tool wants radii. A factor of four in areas and volumes. Fix: check which the input asks for, halve if needed.

Confusing slant height with vertical height. The cone and frustum formulas want the vertical height. If you taped up the sloping face, that is the slant. Fix: height is slant squared minus the radius difference squared, rooted.

Jumping from volume to tonnes without a density. Cubic metres of gravel and of topsoil weigh very different amounts. Fix: volume first, then multiply by a bulk density from the material, not from memory.

How this was checked

The degenerate cases were checked exactly rather than approximately: the frustum with equal radii returns exactly the cylinder volume, the frustum with a zero top radius returns exactly the cone, and the cone returns exactly a third of its cylinder. The sphere at radius 1 returns four thirds pi, its surface exactly four pi, and the worked stockpile above returns pi times 5 over 3 times 156 to full precision.

Related Calculators and Reference

The plane sections these solids stand on are covered by the Area of 2D Shapes Calculator. For real ground rather than ideal solids, the Earthworks Volume Calculator works from an elevation grid, and the Quantity Takeoff Calculator turns volumes into concrete, rebar and formwork quantities. Stockpile tonnage needs a density, for which the Rock Density and Specific Gravity Calculator is the companion step.