Truncated Cone Calculator

Background.

Slice the top off a cone with a cut parallel to its base and the shape that remains — wider circle below, narrower circle above, sloping wall between — is a conical frustum, and it is everywhere: buckets and planters, funnels and hoppers, lampshades, paper cups, volcano cinder cones, concrete pier footings. Its volume has a closed form as old as Egyptian mathematics: V = πh(R² + Rr + r²)/3, with R and r the large and small radii and h the perpendicular height.

The formula's structure rewards a second look. The bracket averages three areas — the big face (πR²), the small face (πr²), and their geometric-mean companion (πRr) — and a third of that average times the height is the volume. Set r = R and it collapses to a cylinder's πR²h; set r = 0 and the full cone's πR²h/3 appears: the frustum interpolates exactly between the two, and the cross-term Rr is what makes the interpolation honest.

That cross-term is also the working answer to the tempting shortcut. Averaging the two radii and computing a cylinder — π((R+r)/2)²h — always underestimates, because volume grows with the square of radius and the wide end contributes more than a linear average credits. For a bucket with R = 5, r = 3, h = 4 the shortcut gives ≈201 against the true ≈205; for steeper tapers the gap widens into real material-ordering errors.

This page computes the right circular frustum — both faces parallel, axis perpendicular — which covers the manufactured world's cups and hoppers; oblique cuts and elliptical sections are outside it, as the scope note says. Slant height, when you need the wall itself, follows from Pythagoras: √(h² + (R−r)²).

What is truncated cone calculator?

A truncated cone (conical frustum) is the solid between two parallel circular faces of radii R and r cut from one cone — equivalently, a cone with its top removed parallel to the base. Its volume is V = πh(R² + Rr + r²)/3 for perpendicular height h: one-third of height times the sum of the two face areas and their geometric-mean term. The formula contains the cylinder (r = R) and the full cone (r = 0) as limiting cases, and its earliest known statement — for square frusta — appears in the Egyptian Moscow Papyrus, c. 1850 BC.

How to use this calculator.

1. 01Measure the two radii — halve the diameters, which is what callipers and tape measures naturally give on rims and bases — and identify R as the larger.
2. 02Measure the perpendicular height h, straight between the two faces — not along the sloping wall; for a vessel, inside height for capacity, outside for displacement.
3. 03Enter all three in the same unit and read the volume in that unit cubed — divide cm³ by 1,000 for litres.
4. 04If only the slant length L along the wall is measurable, recover the height first: h = √(L² − (R−r)²).
5. 05For a partially filled vessel, compute the frustum up to the fill line: the fill-surface radius interpolates linearly, r_fill = r + (R−r)×(fill height ÷ h) for a vessel standing on its narrow end.

The formula.

The clean derivation subtracts cones. Extend the frustum's wall upward to the apex it came from: the full cone has volume πR²H/3 for apex height H, the removed top is a similar cone πr²(H−h)/3, and similarity fixes H through r/R = (H−h)/H, i.e. H = hR/(R−r). Subtracting and simplifying — the difference of cubes R³−r³ factoring as (R−r)(R²+Rr+r²) — cancels the (R−r) and leaves V = πh(R²+Rr+r²)/3, valid even as r → R where the apex retreats to infinity. Calculus agrees in one line: radius varies linearly along the axis, so integrating π·radius² gives the average of the squared radius, and the average of a squared linear function over an interval is exactly (R²+Rr+r²)/3 — the same bracket, now legible as ‘mean cross-section area’. The geometric-mean term πRr is the correction the naive average-radius cylinder omits, and it is why that shortcut always runs low. The engine evaluates the bracket, the product, and the division in Decimal arithmetic, rounding once to twelve significant digits.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06. independence: primary; access: open.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06. independence: secondary-check; access: open.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06. independence: primary; access: open.