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Frustum Volume Calculator

Find the volume of a conical frustum.

Result

513.13 units³

How it is calculated

A conical frustum is a cone with its top cut off by a plane parallel to the base. Its volume is V = ⅓ · π · h · (R² + R·r + r²), where R is the bottom (larger) radius, r is the top radius and h is the perpendicular height between the two circular faces. Use one length unit for all three; the result is in cubic units of that length, shown with two decimals as units³. R and h must be greater than zero, and r may be zero, which turns the shape into a full cone. The tool does not check that R is bigger than r. If r equals R, the formula gives the volume of a cylinder.

Worked example

A bucket has bottom radius R = 5, top radius r = 3 and height h = 10. Compute R² + R·r + r² = 25 + 15 + 9 = 49. Then V = ⅓ × π × 10 × 49 = 513.13. The volume is 513.13 units³. For comparison, a cylinder with radius 5 and the same height would hold 785.40, so the frustum is smaller.

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Common mistakes

  • Entering diameters instead of radii. Halve them first, or the volume comes out four times too large.
  • Using the slant height (along the side) for h. The formula needs the perpendicular height between the two circular faces.
  • Mixing units such as radii in cm and height in metres, which makes the result meaningless.
  • Treating it as a cylinder with the average radius. The average-radius cylinder gives a slightly smaller volume than the true frustum, since the formula includes the R·r term.

About this calculator

A conical frustum is a cone with its tip sliced off by a plane parallel to the base, leaving two circular faces of different radii, R (larger) and r (smaller), separated by height h. Its volume is V = ⅓·π·h·(R² + R·r + r²), a formula derived by integrating circular cross-sections along the axis, which reduces to the standard cone volume formula when r = 0.

This shape appears constantly in engineering and everyday objects: buckets, lampshades, drinking cups, silos, funnels, and traffic cones are all frustums, and calculating their volume is essential for determining material capacity, liquid volume, or the amount of concrete or sheet metal needed to fabricate one.

Manufacturing and civil engineers designing tanks or hoppers, machinists calculating stock material, students working through solid geometry problems, and hobbyists building custom containers all use this formula.

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Comments (1)

  • Editorial team

    For a planter, bucket or lampshade, measure the top and bottom diameters and halve them, since the tool wants radii. If you work in centimetres, divide the result by 1,000 to get litres: radii of 10 and 8 with a height of 20 give about 5,110 cubic centimetres, or 5.11 litres. This is the full volume up to the brim, not a partly filled level. What object are you measuring?

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