Volume of frustum top radius 3, bottom radius 6, height 4 Step by Step

Step by step

The question

Question: a frustum has top radius 3, bottom radius 6 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: 3/6 equals h/h plus 4.

Here h is the small cone's height.

36 = hh + 4

r : R = h : H

Cross multiply, as in the proportion lesson.

3(h + 4) = 6h

Open the bracket.

3h + 12 = 6h

Take away 3h from both sides: 3h.

12 = 3h

Divide by 3: h = 4.

The whole cone is 4 plus 4: H = 8.

h = 4, H = 4 + 4 = 8

2 Big cone

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

6 squared is 36.

One third of 36 times 8 is 96π.

big cone = 13 × π × 6² × 8 = 96π

3 Small cone

The same for the small cone: 12π.

small cone = 13 × π × 3² × 4 = 12π

Careful!

We cannot take away the radii first.

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

Careful: not 13 × π × (6 − 3)² × 4 ✗

4 Take away

96π minus 12π is 84π.

That is about 263.89, to 2 decimal places.

frustum = 96π − 12π = 84π ≈ 263.89

5 Check

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

And 3/6 equals 4/8: the cones are similar.

So it fits!

small cone 37.7 < 263.89 < big cone 301.59

Let's look back.

First: heights of the cones.

Then: big cone.

Then: small cone.

Then: take away.

Last: check.

Answer