Volume of frustum top radius 2, bottom radius 6, height 6 Step by Step

Step by step

The question

Question: a frustum has top radius 2, bottom radius 6 and height 6: find its volume.

A frustum is a cone with its top cut off.

So its volume is the big cone minus the small cone.

1 Heights of the cones

Put the small cone back on top.

The small cone and the big cone are similar: the same shape.

So the radius ratio is the height ratio: 2/6 equals h/h plus 6.

Here h is the small cone's height.

26 = hh + 6

r : R = h : H

Cross multiply, as in the proportion lesson.

2(h + 6) = 6h

Open the bracket.

2h + 12 = 6h

Take away 2h from both sides: 4h.

12 = 4h

Divide by 4: h = 3.

The whole cone is 3 plus 6: H = 9.

h = 3, H = 3 + 6 = 9

2 Big cone

The cone rule: one third times π r squared times the height.

6 squared is 36.

One third of 36 times 9 is 108π.

big cone = 13 × π × 6² × 9 = 108π

3 Small cone

The same for the small cone: 4π.

small cone = 13 × π × 2² × 3 = 4π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (6 − 2)² × 6 ✗

4 Take away

108π minus 4π is 104π.

That is about 326.73, to 2 decimal places.

frustum = 108π − 4π = 104π ≈ 326.73

5 Check

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

And 2/6 equals 3/9: the cones are similar.

So it fits!

small cone 12.57 < 326.73 < big cone 339.29

Let's look back.

First: heights of the cones.

Then: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer