Volume of frustum radii 6 and 9, heights 8 and 12 Step by Step

Step by step

Question: a frustum is a cone with radius 9 and height 12, with a small cone of radius 6 and height 8 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.

9 squared is 81.

One third of 81 times 12 is 324π.

big cone = 13 × π × 9² × 12 = 324π

The same for the small cone: 96π.

small cone = 13 × π × 6² × 8 = 96π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (9 − 6)² × 4 ✗

324π minus 96π is 228π.

That is about 716.28, to 2 decimal places.

frustum = 324π − 96π = 228π ≈ 716.28

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

And 6/9 equals 8/12: the cones are similar.

So it fits!

small cone 301.59 < 716.28 < big cone 1017.88

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 324π − 96π = 228π ≈ 716.28