How to Find the Volume of a Frustum

A frustum is a cone with its top cut off parallel to the base. Its volume is π × height × (top radius² + top radius × bottom radius + bottom radius²) ÷ 3. When the two radii are equal, this reduces to the volume of a cylinder.

Explanation

A frustum is a cone cut by a plane parallel to its base, with the tip taken away: a bucket, a plant pot, a paper cup. Its volume follows from the two radii and the height.

V = \dfrac{\pi h}{3}\left(a^2 + ab + b^2\right)

$V$ — the volume
$a$ — the top radius
$b$ — the bottom radius
$h$ — the height

Swapping $a$ and $b$ leaves the answer unchanged, as it must: turning a bucket upside down does not change what it holds.

Example

With the defaults, a top radius of $a = 3$ , a bottom radius of $b = 5$ and a height of $h = 4$ .

V = \dfrac{\pi \times 4}{3}(3^2 + 3 \times 5 + 5^2) = \dfrac{4\pi}{3} \times 49 \approx 205.2507

Checking the formula

The formula is the large cone minus the small one that was cut away, and both extremes bear it out.

letting the top radius $a$ fall to 0 gives $V = \dfrac{\pi b^2 h}{3}$ , the volume of a cone
setting $a = b$ gives $V = \pi a^2 h$ , the volume of a cylinder

Watch out

The height is the straight distance between the two circles, not the slant length along the side. The slant is what the surface area needs.