Formula and Method for the Truncated Cone (Frustum)
A truncated cone, or frustum, is what remains of a right circular cone after slicing off the top with a cut parallel to the base. It is defined by three measurements: the bottom radius R, the top radius r, and the perpendicular height h between the two circular faces. From these three values, this calculator derives the volume, slant height, and both lateral and total surface area.
How the calculation works
The volume comes from integrating the area of circular cross-sections along the height, which gives V = (πh/3)(R² + Rr + r²) — a generalization of the standard cone volume formula that reduces to it when r = 0. The slant height, the distance along the sloped side, is found with the Pythagorean theorem applied to the height and the difference between the radii: l = √((R − r)² + h²). Once the slant height is known, the lateral (side) surface area is π(R + r)l, and adding the areas of the top and bottom circles, πr² and πR², gives the total surface area A = π(R + r)l + πR² + πr². Enter R, r, and h in the same unit — the calculator reports volume in cubic units and area in square units of whatever you select.
Common mistakes
- Using slant height instead of vertical height: h in the volume formula must be the perpendicular height between the two faces, not the slant length along the side.
- Mixing units: R, r, and h must all be in the same unit before you calculate; convert everything to one unit first.
- Swapping the radii: the formulas are symmetric in R and r for volume and surface area, but keep them consistent with your labeled diagram so the slant height direction makes sense.
Real-world applications
- Buckets, funnels, lampshades, and traffic cones are frustum-shaped, and volume tells you their capacity.
- Concrete footings and retaining wall sections are often poured as frustums, where volume determines material quantity.
- Sheet-metal and fabrication work uses the lateral surface area to calculate how much flat material is needed to wrap the tapered side.
- Silos, hoppers, and industrial chutes commonly use a frustum transition section between two different pipe or duct diameters.