Step by step
The question
Question: a frustum is a cone with radius 6 and height 12, 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
A cone is one third of the cylinder around it: ⅓ × π × r² × h.
The big cone has r = 6 and h = 12.
6² means 6 times 6.
That is 36.
⅓ of 36 means 36 divided by 3.
We can multiply in any order.
So we do that first: it keeps the numbers small.
36 divided by 3 is 12.
So ⅓ × 36 becomes 12.
So big cone = π × 12 × 12.
12 times 12 is 144.
So the big cone is 144π.
2 Small cone
The same rule for the small cone: r = 2 and h = 4.
2² means 2 times 2.
That is 4.
None of the numbers divides by 3.
So we multiply and keep the 3 at the bottom: 4 times 4 is 16.
So the small cone is 16π over 3.
Careful!
We cannot take away the radii first.
We work out the two whole cones, then take away.
3 Take away
Now the big cone minus the small cone.
To take away, both need the same bottom: 3.
Times 3 on top and at the bottom changes nothing.
144 times 3 is 432.
So 144π is 432π over 3.
So we take away 16π over 3 from 432π over 3.
Thirds minus thirds: take away the tops, the bottom stays 3.
432 minus 16 is 416.
So the frustum is 416π over 3.
We keep π in the answer.
That is the exact answer.
4 About how much
π is a special number for circles: the way round is about 3.14 times the way across.
To see about how much, put 3.14 for π.
416 times 3.14 is 1306.24.
1306.24 divided by 3 is about 435.41.
So the volume is about 435.41.
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 4/12: the cones have the same shape.
So it fits!
Let's look back.
First: big cone.
Then: small cone.
Then: take away.
Then: about how much.
Last: check.