Volume of frustum top radius 3, bottom radius 9, height 12 Step by Step

Step by step

The question

Question: a frustum has top radius 3, bottom radius 9 and height 12: 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: 3/9 equals h/h plus 12.

Here h is the small cone's height.

39 = hh + 12

r : R = h : H

Cross multiply, as in the proportion lesson.

3(h + 12) = 9h

Open the bracket.

3h + 36 = 9h

Take away 3h from both sides: 6h.

36 = 6h

Divide by 6: h = 6.

The whole cone is 6 plus 12: H = 18.

h = 6, H = 6 + 12 = 18

2 Big cone

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

9 squared is 81.

One third of 81 times 18 is 486π.

big cone = 13 × π × 9² × 18 = 486π

3 Small cone

The same for the small cone: 18π.

small cone = 13 × π × 3² × 6 = 18π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (9 − 3)² × 12 ✗

4 Take away

486π minus 18π is 468π.

That is about 1470.27, to 2 decimal places.

frustum = 486π − 18π = 468π ≈ 1470.27

5 Check

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

And 3/9 equals 6/18: the cones are similar.

So it fits!

small cone 56.55 < 1470.27 < big cone 1526.81

Let's look back.

First: heights of the cones.

Then: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer