A conical frustum — the shape you get by slicing the tip off a cone — is fully defined by three numbers: the radius of its wider end, the radius of its narrower end, and its height. From those three values alone, you can derive the slant height, the lateral surface area, the total surface area, and the volume. This calculator computes all of them instantly, and reduces cleanly to the standard cone formulas when one radius is zero.

How the Frustum Calculator works

The calculator starts from the volume formula V = (1/3)πh(R² + Rr + r²), a generalization of the cone volume formula that averages the contribution of both circular ends. It then finds the slant height with the Pythagorean theorem, l = √((R − r)² + h²), since the difference between the two radii and the height form a right triangle along the frustum's sloped side. The lateral surface area is π(R + r)l, and adding the two circular end areas (πR² and πr²) gives the total surface area.

Inputs and what they mean

Enter the major radius (R) and minor radius (r) — the radii of the two parallel circular ends — and the height (h), the perpendicular distance between them. All three must be non-negative numbers in the same unit; the calculator does not convert between units, so pick inches, feet, centimeters, or meters up front and stay consistent. Every formula here is symmetric in R and r, so entering the smaller radius as R and the larger as r still produces correct results — the calculator will simply note that the frustum is flipped relative to the major/minor labels.

Limits and edge cases

Setting the minor radius r to 0 turns every frustum formula into the matching cone formula exactly: volume becomes (1/3)πR²h, slant height becomes √(R² + h²), and total surface area becomes πRl + πR² — use the Cone Calculator directly if you always work with a pointed cone. Setting both radii equal (R = r) turns the frustum into a cylinder, since the slant height then equals the height and the lateral formula collapses to 2πRh. This calculator requires a positive height and non-negative radii that aren't both zero; it does not model oblique (slanted-axis) frustums.