Volume of frustum top radius 2 cm, bottom radius 4 cm, height 5 cm Step by Step

Step by step

The question

Question: a frustum has top radius 2 cm, bottom radius 4 cm and height 5 cm: 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: 2/4 equals h/h plus 5.

Here h is the small cone's height.

24 = hh + 5

r : R = h : H

Cross multiply, as in the proportion lesson.

2(h + 5) = 4h

Open the bracket.

2h + 10 = 4h

Take away 2h from both sides: 2h.

10 = 2h

Divide by 2: h = 5.

The whole cone is 5 plus 5: H = 10.

h = 5 cm, H = 5 + 5 = 10 cm

2 Big cone

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

4 squared is 16.

One third of 16 times 10 is 160π over 3.

big cone = 13 × π × 4² × 10 = 160π3

3 Small cone

The same for the small cone: 20π over 3.

small cone = 13 × π × 2² × 5 = 20π3

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (4 − 2)² × 5 ✗

4 Take away

160π over 3 minus 20π over 3 is 140π over 3.

That is about 146.61 cm³, to 2 decimal places.

frustum = 160π3 − 20π3 = 140π3 ≈ 146.61 cm³

5 Check

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

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

So it fits!

small cone 20.94 < 146.61 < big cone 167.55

Let's look back.

First: heights of the cones.

Then: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer