Volume of a Cone

Calculates the volume of a cone as π × radius² × height ÷ 3, exactly one third of the cylinder with the same base and height.

Explanation

A cone rises from a circular base to a single point, the apex. Its volume is exactly one third of the cylinder that shares its base and its height.

V = \dfrac{1}{3} \pi r^2 h

$V$ — the volume
$r$ — the radius of the base
$h$ — the height, the perpendicular distance from the base to the apex
$\pi$ — pi

Example

With the defaults, a radius of $r = 3$ and a height of $h = 6$ . The base area is $\pi \times 3^2 = 9\pi$ , so

V = \dfrac{1}{3} \times 9\pi \times 6 = 18\pi \approx 56.5487

The volume is about 56.5487. A cylinder with the same base and height would hold $9\pi \times 6 = 54\pi$ , and the cone is precisely a third of that.

Watch out

The height is the perpendicular distance from the base to the apex, not the slant height, which is the sloping line from the apex down to the rim. If you know the slant height $\ell$ but not $h$ , recover the height with $h = \sqrt{\ell^2 - r^2}$ before using the formula. Forget to divide by 3 and you have computed the cylinder instead. Pyramids obey the same rule: base area times height, divided by three. Both the radius and the height must be positive.