Volume of frustum radii 4 and 8, heights 5 and 10 Step by Step

Step by step

Question: a frustum is a cone with radius 8 and height 10, with a small cone of radius 4 and height 5 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.

8 squared is 64.

One third of 64 times 10 is 640π over 3.

big cone = 13 × π × 8² × 10 = 640π3

The same for the small cone: 80π over 3.

small cone = 13 × π × 4² × 5 = 80π3

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (8 − 4)² × 5 ✗

640π over 3 minus 80π over 3 is 560π over 3.

That is about 586.43, to 2 decimal places.

frustum = 640π3 − 80π3 = 560π3 ≈ 586.43

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

And 4/8 equals 5/10: the cones are similar.

So it fits!

small cone 83.78 < 586.43 < big cone 670.21

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 640π3 − 80π3 = 560π3 ≈ 586.43