Volume of frustum radii 2 and 4, heights 3 and 6 Step by Step

Step by step

Question: a frustum is a cone with radius 4 and height 6, 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.

4 squared is 16.

One third of 16 times 6 is 32π.

big cone = 13 × π × 4² × 6 = 32π

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 × (4 − 2)² × 3 ✗

32π minus 4π is 28π.

That is about 87.96, to 2 decimal places.

frustum = 32π − 4π = 28π ≈ 87.96

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

And 2/4 equals 3/6: the cones are similar.

So it fits!

small cone 12.57 < 87.96 < big cone 100.53

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 32π − 4π = 28π ≈ 87.96