Volume of frustum radii 3 and 6, heights 4 and 8 Step by Step

Step by step

Question: a frustum is a cone with radius 6 and height 8, with a small cone of radius 3 and height 4 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 8 is 96π.

big cone = 13 × π × 6² × 8 = 96π

The same for the small cone: 12π.

small cone = 13 × π × 3² × 4 = 12π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (6 − 3)² × 4 ✗

96π minus 12π is 84π.

That is about 263.89, to 2 decimal places.

frustum = 96π − 12π = 84π ≈ 263.89

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

And 3/6 equals 4/8: the cones are similar.

So it fits!

small cone 37.7 < 263.89 < big cone 301.59

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 96π − 12π = 84π ≈ 263.89