Volume of frustum top radius 1, bottom radius 3, height 4 Step by Step

Step by step

The question

Question: a frustum has top radius 1, bottom radius 3 and height 4: 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: 1/3 equals h/h plus 4.

Here h is the small cone's height.

13 = hh + 4

r : R = h : H

Cross multiply, as in the proportion lesson.

1(h + 4) = 3h

Open the bracket.

1h + 4 = 3h

Take away 1h from both sides: 2h.

4 = 2h

Divide by 2: h = 2.

The whole cone is 2 plus 4: H = 6.

h = 2, H = 2 + 4 = 6

2 Big cone

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

3 squared is 9.

One third of 9 times 6 is 18π.

big cone = 13 × π × 3² × 6 = 18π

3 Small cone

The same for the small cone: 2π over 3.

small cone = 13 × π × 1² × 2 = 2π3

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (3 − 1)² × 4 ✗

4 Take away

18π minus 2π over 3 is 52π over 3.

That is about 54.45, to 2 decimal places.

frustum = 18π − 2π3 = 52π3 ≈ 54.45

5 Check

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

And 1/3 equals 2/6: the cones are similar.

So it fits!

small cone 2.09 < 54.45 < big cone 56.55

Let's look back.

First: heights of the cones.

Then: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer