Step by step
The question
Question: find the volume of a sphere (a ball) with radius 6.
1 Use the rule
Put the ball in a cylinder just as wide and just as high.
The ball fills exactly two thirds of that cylinder.
Archimedes found this, long ago.
π is a special number for circles: the way round is about 3.14 times the way across.
The cylinder is as high as the ball is wide: 2r.
So the cylinder is π × r² × 2r.
r² times r is r³: r times r times r.
So the cylinder is 2 × π × r³.
Careful: r³ is r times r times r.
It is not 3 times r.
The ball is two thirds of the cylinder: two thirds of 2 × π × r³.
Two thirds times 2 is 4 thirds.
So the volume is V = 4/3 × π × r³.
Now put the numbers in: r = 6.
6³ means 6 times 6 times 6.
That is 216.
So V = 4/3 × π × 216.
4/3 means times 4, divided by 3.
We can multiply in any order.
So we divide 216 by 3 first: it keeps the numbers small.
216 divided by 3 is 72.
So V = 4 × π × 72.
4 times 72 is 288.
So V = 288π.
We keep π in the answer.
That is the exact answer.
2 About how much
To see about how much, put 3.14 for π.
288 times 3.14 is 904.32.
So the volume is about 904.32 cubic units.
Let's look back.
First: use the rule.
Last: about how much.