Formula

Volume of a frustum of a cone

Mensuration • Solids • Frustum

Formula

V = \frac{π h}{3} (R^{2} + R r + r^{2})

A frustum of a cone with parallel circular ends of radii

R

and

r

and perpendicular height

h

between them has volume

\frac{π h}{3} (R^{2} + R r + r^{2})

How to use it

1Halve both stated diameters first; the formula takes radii, and questions describe drinking glasses and buckets by their diameters.
2The $h$ is the height of the frustum itself, not of the cone it was cut from.
3As a check, restore the full cone by similarity and subtract the small cone removed; the two routes must agree.

A bucket is a frustum of height

15

cm with end radii

10

cm and

5

cm. Find its capacity in terms of

π

Step-by-step

1
$R^{2} + R r + r^{2} = 100 + 50 + 25 = 175$
Why: Evaluate the bracket before touching the rest.
2
$V = \frac{π \times 15}{3} \times 175 = 875 π$ cm $^{3}$

Answer

875 π

^{3}

Watch out

The bracket is

R^{2} + R r + r^{2}

and not

(R + r)^{2}

R^{2} + r^{2}

; and the leading factor is

\frac{1}{3}

, inherited from the cone. Setting

r = 0

must return the cone formula, which is the quickest way to check a remembered version.

Builds on

Also known as

volume of a frustum, frustum of a cone, bucket volume, truncated cone