Worked examples
The question
Question: a frustum is a cone with radius 6 and height 8, with a small cone of radius 3 and height 4 cut off the top: 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 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
2 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 ✗
3 Take away
96π minus 12π is 84π.
That is about 263.89, to 2 decimal places.
frustum = 96π − 12π = 84π ≈ 263.89
4 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: big cone.
Then: small cone.
Then: take away.
Last: check.
Answer frustum = 96π − 12π = 84π ≈ 263.89
▶ Watch it solved
The question
Question: a frustum is a cone with radius 3 and height 6, with a small cone of radius 2 and height 4 cut off the top: 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 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
2 Small cone
The same for the small cone: 16π over 3.
small cone = 13 × π × 2² × 4 = 16π3
Careful!
We cannot take away the radii first.
We work out the two whole cones, then take away.
Careful: not 13 × π × (3 − 2)² × 2 ✗
3 Take away
18π minus 16π over 3 is 38π over 3.
That is about 39.79, to 2 decimal places.
frustum = 18π − 16π3 = 38π3 ≈ 39.79
4 Check
Let's check: the frustum is smaller than the big cone and bigger than the small cone, as it should be.
And 2/3 equals 4/6: the cones are similar.
So it fits!
small cone 16.76 < 39.79 < big cone 56.55
Let's look back.
First: big cone.
Then: small cone.
Then: take away.
Last: check.
Answer frustum = 18π − 16π3 = 38π3 ≈ 39.79
▶ Watch it solved
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 frustum = 108π − 4π = 104π ≈ 326.73
▶ Watch it solved
