A frustum is what you get when you cut the top off a cone with a horizontal slice. It has two circular faces — a larger bottom circle with radius R and a smaller top circle with radius r — joined by a curved lateral surface, with a vertical height h between them. The shape appears constantly in real-world engineering: hoppers, grain bins, buckets, drinking cups, lampshades, cooling towers, and funnel-shaped containers are all frustum-like solids. Knowing their volume is essential for capacity planning, materials estimation, and fluid calculations.
This calculator computes the volume of any frustum of a right circular cone. It also works in reverse: give it the volume plus two of the three dimensions and it solves for the missing one. All computation runs locally in your browser using JavaScript — no data leaves your device.
How it works
The volume formula for a frustum is derived by subtracting the volume of the removed top cone from the original full cone. The result is a clean closed-form expression:
V = (pi x h / 3) x (R-squared + R x r + r-squared)
where R is the larger bottom radius, r is the smaller top radius, and h is the perpendicular height (not the slant height). The term inside the parentheses — R-squared plus R times r plus r-squared — is sometimes called the prismatoid sum. It accounts for the three distinct cross-sectional contributions: the bottom circle area, the geometric mean area, and the top circle area.
Solve-for modes
The calculator exposes four solve-for modes:
| Mode | Given | Solves for |
|---|---|---|
| Volume | R, r, h | V |
| Height | V, R, r | h = 3V / (pi x (R-squared + Rr + r-squared)) |
| Top radius | V, R, h | r — via quadratic formula |
| Bottom radius | V, r, h | R — via quadratic formula |
For the reverse-radius modes the formula becomes a quadratic in the unknown radius. For example, solving for r: expand V = (pi x h / 3) x (R-squared + R x r + r-squared) and rearrange to get r-squared plus R x r plus (R-squared minus 3V/(pi x h)) = 0. The positive root of that quadratic is the answer. The calculator discards any negative root automatically.
Worked example
A steel hopper has a bottom radius of 5 cm, a top radius of 3 cm and a height of 8 cm. What is its volume?
Step 1 — compute the prismatoid sum:
R-squared + R x r + r-squared = 25 + 15 + 9 = 49
Step 2 — apply the formula:
V = (pi x 8 / 3) x 49 = (8.3776) x 49 = about 410.50 cm-cubed
In litres that is roughly 0.411 litres — a little less than a standard coffee mug. You can verify by entering R = 5, r = 3, h = 8 in the calculator above.
| R (cm) | r (cm) | h (cm) | Volume (cm-cubed) |
|---|---|---|---|
| 5 | 3 | 8 | 410.50 |
| 10 | 5 | 12 | 2199.11 |
| 8 | 0 | 10 | 670.21 |
| 4 | 4 | 6 | 301.59 |
Row 3 (r = 0) is a plain cone; row 4 (R = r) is a cylinder — both are handled correctly by the same formula.
Formula note
When R equals r the frustum becomes a cylinder and the formula gives V = pi x r-squared x h (correct, because R-squared + R x r + r-squared = 3r-squared, and (pi x h / 3) x 3r-squared = pi x r-squared x h). When r equals 0 it becomes a full cone and V = (pi x h / 3) x R-squared — again correct. So this single formula covers cones, cylinders and all frustums in between.