Frustum Volume Calculator

A frustum is a portion of a cone or pyramid lying between two parallel planes that cut the solid. When you slice a right circular cone horizontally below its apex, you produce a frustum with a large circular base of radius R, a small circular top of radius r, and a vertical height h. The same concept applies to pyramids, where the bases are polygons — most commonly squares.

For a circular frustum, the volume formula is derived by computing the volume of the full cone that would have existed before truncation, then subtracting the small cone that was removed. This yields V = (πh/3)(R² + Rr + r²). When r = 0, the formula collapses to the standard cone volume V = πR²h/3. When r = R, it reduces to the cylinder formula V = πR²h. For a square-base frustum, the formula generalises to V = (h/3)(A₁ + A₂ + √(A₁A₂)), where A₁ = R² and A₂ = r² are the base areas. The slant height l = √((R−r)² + h²) describes the distance along the tapered side, used to calculate the lateral surface area πl(R+r) for circular frustums.

Frustum calculations are essential across many professional fields. Civil engineers calculate the volume of earthwork cuts and fills with frustum-shaped cross-sections. Structural designers compute the material volume of tapered columns, hoppers, and silos. In manufacturing, frustum geometry governs the material removed during turned machining operations. Packaging designers use it for tapered cups, buckets, and funnels, where accurate volume is critical for product labelling. Students use the formula extensively in university entrance examinations and physics problems involving mass and centre of gravity of truncated solids.

This calculator assumes the frustum is a right frustum — the two parallel bases are perpendicular to the central axis, and the apex of the original cone or pyramid lies directly above the centroid of the base. Oblique frustums, where the cut is not parallel to the base, require more complex integration-based methods. For circular frustums, both bases are assumed to be perfect circles; elliptical or irregular cross-sections are not supported. The square frustum variant assumes both bases are squares aligned on the same axis — rectangular or polygonal frustums with unequal side lengths need adapted base-area formulas. All inputs must be positive real numbers; negative or zero radii and height values are geometrically meaningless. In engineering applications, always verify units are consistent (all metres, all centimetres, etc.) before interpreting the output. Material volumes calculated here do not account for wall thickness in hollow frustums such as funnels or tapered pipes.