Volume of frustum radii 3 and 4, heights 9 and 12 Step by Step

Step by step

Question: a frustum is a cone with radius 4 and height 12, with a small cone of radius 3 and height 9 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.

4 squared is 16.

One third of 16 times 12 is 64π.

big cone = 13 × π × 4² × 12 = 64π

The same for the small cone: 27π.

small cone = 13 × π × 3² × 9 = 27π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (4 − 3)² × 3 ✗

64π minus 27π is 37π.

That is about 116.24, to 2 decimal places.

frustum = 64π − 27π = 37π ≈ 116.24

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

And 3/4 equals 9/12: the cones are similar.

So it fits!

small cone 84.82 < 116.24 < big cone 201.06

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 64π − 27π = 37π ≈ 116.24