What is a truncated cone?
A truncated cone — also called a conical frustum — is the solid you get when you slice the pointed top off a cone with a cut parallel to its base. It has 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 buckets, lampshades, drinking cups and flower pots. This calculator returns the volume, slant height, lateral surface area and total surface area in one step.
How to use this calculator
Enter the bottom radius R, the top radius r, and the perpendicular height h, all measured in the same units (cm, m, inches, etc.). The result gives the volume in cubic units and every surface measurement in square units. If you only know the diameters, divide each by two first. Setting \(r = 0\) turns the frustum back into a full cone.
The formulas explained
The volume is $$V = \tfrac{1}{3}\cdot\pi\cdot h\cdot\left(R^{2} + Rr + r^{2}\right),$$ an average of the two circular areas weighted by the cross-term \(Rr\). The slant height — the diagonal distance along the slanted side — is $$\ell = \sqrt{(R - r)^{2} + h^{2}}$$ from the Pythagorean theorem. The curved side, or lateral area, is $$A = \pi(R + r)\cdot\ell.$$ Adding the two flat circles (\(\pi R^{2}\) and \(\pi r^{2}\)) gives the total surface area.
Worked example
For \(R = 5\), \(r = 3\), \(h = 8\): $$V = \tfrac{1}{3}\cdot\pi\cdot 8\cdot(25 + 15 + 9) = \tfrac{1}{3}\cdot\pi\cdot 8\cdot 49 \approx 410.50 \text{ cubic units}.$$ The slant height $$\ell = \sqrt{(5-3)^{2} + 8^{2}} = \sqrt{68} \approx 8.246.$$ Lateral area \(= \pi\cdot(5+3)\cdot 8.246 \approx 207.24\) square units, and total area adds \(\pi\cdot 25 + \pi\cdot 9 \approx 314.06\).
FAQ
Is height the same as slant height? No. Height \(h\) is the straight vertical distance between the two circles; slant height \(\ell\) runs along the angled surface and is always longer.
Does it matter which radius is larger? No — the formulas are symmetric in \(R\) and \(r\), so swapping them gives the same volume and areas.
What units does it use? Any, as long as all three inputs share the same unit. Volume comes out cubed and areas squared.
