What is a conical frustum?
A conical frustum is the solid you get when you slice the top off a right circular cone with a cut parallel to its base. The result is a truncated cone with two circular faces: a larger bottom of radius R and a smaller top of radius r, separated by a vertical height h. Everyday examples include drinking cups, lampshades, buckets, and flower pots.
How to use this calculator
Enter the bottom radius (\(R\)), the top radius (\(r\)), and the perpendicular height (\(h\)) in any consistent unit. The calculator returns the volume in cubic units, the slant height, the lateral (curved) surface area, the area of each circular face, and the total surface area. If the top radius equals zero, the frustum becomes a full cone; if \(R\) equals \(r\), it becomes a cylinder.
The formulas explained
The volume uses the average-of-cross-sections rule:
$$V = \frac{1}{3}\pi\,\text{h}\left(\text{R}^{2} + \text{R}\,\text{r} + \text{r}^{2}\right)$$The slant height is the straight-line distance along the sloping side, found with the Pythagorean theorem:
$$\ell = \sqrt{\left(\text{R} - \text{r}\right)^{2} + \text{h}^{2}}$$The curved surface wrapping the frustum has lateral area \(A_L = \pi\left(\text{R} + \text{r}\right)\ell\). Adding the two circular ends (\(\pi\,\text{R}^{2}\) and \(\pi\,\text{r}^{2}\)) gives the total surface area.
Worked example
For \(R = 5\), \(r = 3\), \(h = 8\):
$$V = \frac{1}{3}\cdot\pi\cdot 8\cdot\left(25 + 15 + 9\right) = \frac{1}{3}\cdot\pi\cdot 8\cdot 49 \approx 410.50 \text{ cubic units}$$Slant height \(= \sqrt{\left(5-3\right)^{2} + 8^{2}} = \sqrt{4 + 64} = \sqrt{68} \approx 8.246\). Lateral area \(= \pi\cdot\left(5+3\right)\cdot 8.246 \approx 207.23\) square units.
FAQ
Is h the slant or vertical height? Enter the vertical (perpendicular) height. The slant height is computed for you.
What units does it use? Any units, as long as \(R\), \(r\), and \(h\) share the same one. Volume comes out cubed, areas squared.
Does the order of R and r matter? No — the formulas are symmetric, so swapping the two radii gives the same volume and surface area.
