Volume of a cone: formula, calculator and worked example

What the symbols mean

$V$: the volume
$r$: the radius of the circular base
$h$: the perpendicular height, from the base straight up to the tip (not the slant height)

When to use it

Use it for ice-cream cones, funnels, piles of sand or gravel, traffic cones and conical roofs. If you are given the slant height l, find h first with Pythagoras: $h = √(l² − r²)$ .

Worked example

A cone has a base radius of 3 cm and a height of 4 cm. What is its volume? (r = 3, h = 4)

Start from the formula.
$V = \frac{1}{3} π r^{2} h$
Substitute $r$ = $3$ , $h$ = $4$ and work it out.
$V = \frac{1}{3} π \cdot 3^{2} \cdot 4$
$= \frac{1}{3} π \cdot 9 \cdot 4$
$\approx 37.6991$

Answer V ≈ 37.70 cm³

Why it works

Any cone or pyramid holds exactly a third of the prism with the same base and height: fill the cone with water three times to fill the cylinder. Calculus proves it by adding up thin circular slices.

Common mistakes

Using the slant height instead of the perpendicular height.
Forgetting the ⅓, which gives the volume of the whole cylinder.

Volume of a cone

Calculator

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What the symbols mean

When to use it

Worked example

Why it works

Common mistakes