Volume of frustum radii 2 cm and 5 cm, heights 4 cm and 10 cm Step by Step

Step by step

Question: a frustum is a cone with radius 5 cm and height 10 cm, with a small cone of radius 2 cm and height 4 cm 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.

5 squared is 25.

One third of 25 times 10 is 250π over 3.

big cone = 13 × π × 5² × 10 = 250π3

The same for the small cone: 16π over 3.

small cone = 13 × π × 2² × 4 = 16π3

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (5 − 2)² × 6 ✗

250π over 3 minus 16π over 3 is 78π.

That is about 245.04 cm³, to 2 decimal places.

frustum = 250π3 − 16π3 = 78π ≈ 245.04 cm³

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

And 2/5 equals 4/10: the cones are similar.

So it fits!

small cone 16.76 < 245.04 < big cone 261.8

Let's look back.

First: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer

frustum = 250π3 − 16π3 = 78π ≈ 245.04 cm³