Volume of frustum radii 1 and 3, heights 2 and 6 Step by Step

Step by step

Question: a frustum is a cone with radius 3 and height 6, with a small cone of radius 1 and height 2 cut off the top: find its volume.

A frustum is a cone with its top cut off.

So its volume is the big cone minus the small cone.

The cone rule: one third times π r squared times the height.

3 squared is 9.

One third of 9 times 6 is 18π.

big cone = 13 × π × 3² × 6 = 18π

The same for the small cone: 2π over 3.

small cone = 13 × π × 1² × 2 = 2π3

Careful!

We cannot take away the radii first.

We work out the two whole cones, then take away.

Careful: not 13 × π × (3 − 1)² × 4 ✗

18π minus 2π over 3 is 52π over 3.

That is about 54.45, to 2 decimal places.

frustum = 18π − 2π3 = 52π3 ≈ 54.45

Let's check: the frustum is smaller than the big cone and bigger than the small cone, as it should be.

And 1/3 equals 2/6: the cones are similar.

So it fits!

small cone 2.09 < 54.45 < big cone 56.55

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 18π − 2π3 = 52π3 ≈ 54.45