Volume of frustum radii 4 and 10, heights 6 and 15 Step by Step

Step by step

Question: a frustum is a cone with radius 10 and height 15, with a small cone of radius 4 and height 6 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.

10 squared is 100.

One third of 100 times 15 is 500π.

big cone = 13 × π × 10² × 15 = 500π

The same for the small cone: 32π.

small cone = 13 × π × 4² × 6 = 32π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (10 − 4)² × 9 ✗

500π minus 32π is 468π.

That is about 1470.27, to 2 decimal places.

frustum = 500π − 32π = 468π ≈ 1470.27

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

And 4/10 equals 6/15: the cones are similar.

So it fits!

small cone 100.53 < 1470.27 < big cone 1570.8

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 500π − 32π = 468π ≈ 1470.27