What this calculator does
A truncated cone, also called a frustum, is what is left of a full cone once its pointed top has been sliced off with a cut parallel to the base. Buckets, lampshades, plant pots and traffic cones with a flat top are all truncated cone shapes, with a wider base radius, a narrower top radius, and a height measured straight up between the two flat faces.
Because a truncated cone has two different radii rather than one, its volume and surface area need a different truncated cone formula from a full cone, one that folds both radii into the calculation. This calculator takes the bottom radius, top radius and height and returns the volume, slant height and full surface area breakdown.
The formula
Volume is V = (πh/3)(R² + Rr + r²), where R is the bottom radius, r is the top radius and h is the height. The slant height is the straight-line distance along the sloped side, l = √(h² + (R − r)²), found from the height and the difference between the two radii. Total surface area adds the curved lateral surface, π(R + r)l, to the two flat circular ends, πR² and πr².
| Term | Meaning |
|---|---|
| R (bottom radius) | The radius of the larger, base circle of the frustum. |
| r (top radius) | The radius of the smaller, top circle of the frustum. |
| h (height) | The perpendicular distance between the two flat circular faces, not the slant length. |
| Slant height (l) | The distance measured along the sloped side of the frustum, from the edge of the base to the edge of the top. |
The inputs explained
| Field | What to enter |
|---|---|
| Bottom radius | The radius of the wider, bottom face. |
| Top radius | The radius of the narrower, top face. Set this to zero to get the volume and surface area of a full cone instead. |
| Height | The vertical height between the two flat faces, measured straight up, not along the slope. |
When to use it
Estimating the capacity of a bucket or planter
A bucket or plant pot is a truncated cone in almost every case; the volume formula gives its capacity directly from the two radii and height, without needing to fill it and measure.
Working out material for a lampshade or duct transition
The lateral surface area is the flat pattern, or truncated cone net, needed before it is curved into shape, which is the figure to use when estimating material for a shade, funnel or ducting transition piece.
Checking a design against a full cone
Setting the top radius to zero turns the same calculation into a full cone, which is a useful way to sanity-check the formula, or to compare how much volume is lost by truncating a cone at a given height.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How volume and surface area change as the top radius grows
A fixed 8-unit bottom radius and 10-unit height, as the top radius increases from a full cone (top radius 0) toward a cylinder (top radius equal to the bottom).
| Top radius | Volume | Slant height | Total surface area |
|---|---|---|---|
| 0 | 670.206 | 12.806 | 522.918 |
| 2 | 879.646 | 11.662 | 579.998 |
| 4 | 1,172.86 | 10.770 | 657.359 |
| 6 | 1,549.85 | 10.198 | 762.692 |
| 8 | 2,010.62 | 10.000 | 904.779 |
How volume changes with height at fixed radii
The same two radii, across a range of heights.
| Height | Volume | Slant height |
|---|---|---|
| 4 | 469.145 | 5.657 |
| 6 | 703.717 | 7.211 |
| 8 | 938.289 | 8.944 |
| 10 | 1,172.86 | 10.770 |
| 15 | 1,759.29 | 15.524 |
| 20 | 2,345.72 | 20.396 |
Questions
What is the formula for the volume of a truncated cone?
V = (πh/3)(R² + Rr + r²), where R and r are the bottom and top radii and h is the height between the two flat faces.
What is a truncated cone net?
The flat, two-dimensional pattern that folds up into the truncated cone shape: a curved sector for the lateral surface, sized using the slant height, plus two circles for the top and bottom faces.
How is a truncated cone different from a full cone?
A full cone comes to a single point at the top, equivalent to a truncated cone with a top radius of zero. A truncated cone instead has a flat circular top of its own, cut parallel to the base.
Why does the formula use height rather than slant height for volume?
Volume depends on how much vertical space the shape occupies, which is governed by the perpendicular height, not the length of the sloped side. Slant height only enters the calculation once surface area, rather than volume, is being worked out.
For a full cone with a single radius, see the cone calculator. For a cylinder, see the cylinder calculator.