Volume of frustum radii 2 and 6, heights 3 and 9 Step by Step

Step by step

Question: a frustum is a cone with radius 6 and height 9, with a small cone of radius 2 and height 3 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.

6 squared is 36.

One third of 36 times 9 is 108π.

big cone = 13 × π × 6² × 9 = 108π

The same for the small cone: 4π.

small cone = 13 × π × 2² × 3 = 4π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (6 − 2)² × 6 ✗

108π minus 4π is 104π.

That is about 326.73, to 2 decimal places.

frustum = 108π − 4π = 104π ≈ 326.73

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

And 2/6 equals 3/9: the cones are similar.

So it fits!

small cone 12.57 < 326.73 < big cone 339.29

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 108π − 4π = 104π ≈ 326.73