πh ÷ 3 × (R² + Rr + r²)

Frustum calculator

Two radii and a height give the volume of a truncated cone, its slant height, its curved surface and the litres it holds.

Short answer: a frustum with bottom radius 3, top radius 1 and height 4 has volume 54.45 cubic units — 54,454 litres if those units are metres — a slant height of √20 ≈ 4.472 and a curved surface of about 56.2 square units.

Why the cylinder formula is wrong here

πr²h assumes the same radius all the way up. A tapered vessel holds less than its bottom radius suggests, and the extra Rr term in the frustum formula is exactly the correction. A 10-litre pot guessed as a cylinder can come out closer to 7.

Worked example

A bucket 30 cm across the bottom, 24 cm across the top and 35 cm tall: R = 0.15 m, r = 0.12 m, h = 0.35 m gives V = 0.0199 m³ ≈ 19.9 litres. Adding the two circular faces gives the sheet metal needed, which is why the surface areas are printed alongside.

Frustum questions

What is a frustum?

A cone with the top sliced off parallel to the base, leaving two circular faces. A bucket, a plant pot and a lampshade are all frustums.

What is the volume formula?

V = πh ÷ 3 × (R² + Rr + r²), where R is the bottom radius and r the top. Those three squared terms are what make it more than πr²h — using the cylinder formula overestimates a tapered vessel.

How is the slant height found?

Pythagoras on the height and the radius difference: √(h² + (R − r)²). It is needed for the curved surface area, and it is not the same as the height unless the sides are vertical.

Does it handle equal radii?

Yes — when both radii match, the result reduces exactly to a cylinder: πr²h, with the slant height equal to the height. That case is a good way to check the answer looks sane.

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