Formula and Method for Truncated Cone (Frustum) Volume
A truncated cone, or frustum, is what remains of a cone after slicing off the top with a cut parallel to the base — leaving two parallel circular faces of different sizes connected by a slanted lateral surface. Its volume is V = (πh/3)(R² + Rr + r²), where R is the larger (bottom) radius, r is the smaller (top) radius, and h is the perpendicular height between the two circular faces. This calculator also derives the slant height and both the lateral and total surface area from the same three measurements.
How the calculation works
Enter the bottom radius R, the top radius r, and the perpendicular height h, all in the same unit. The volume formula V = (πh/3)(R² + Rr + r²) comes from integrating the area of circular cross-sections along the height, and it correctly reduces to the ordinary cone formula πR²h/3 when r = 0, and to a cylinder's πR²h when R = r. The slant height, l = √(h² + (R − r)²), follows from the Pythagorean theorem applied to the right triangle formed by the height, the radius difference (R − r), and the slanted side. Once l is known, the lateral (side) surface area is π(R + r)l, and adding the two circular base areas (πR² and πr²) gives the total surface area A = π(R² + r²) + π(R + r)l.
Common mistakes
- Mixing up R and r: R must be the larger radius and r the smaller one — swapping them changes the sign of (R − r) inside the slant-height formula, though the volume formula itself is symmetric in R and r.
- Using diameter instead of radius: if you measured across the full circle, divide by 2 before entering the value — using diameter as radius roughly quadruples the computed volume.
- Confusing height with slant height: h is the straight vertical (perpendicular) distance between the two faces, not the length along the slanted side; using the slant length in the volume formula overstates the result.
Real-world applications
- Buckets, lampshades, plant pots, and traffic cones are everyday frustum shapes where volume determines capacity.
- Hoppers, funnels, and silos in manufacturing and agriculture use frustum volume to size storage and flow-through capacity.
- Concrete footings, retaining walls, and tapered columns in construction use the formula to estimate material quantities.
- Lampshade fabric, cone-shaped packaging, and tapered vessel design use the lateral surface area to estimate material needed.