Worked examples
The question
Question: find the volume of a sphere (a ball) with radius 3.
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.
Worked out, that gives the rule for a ball.
The rule has r³: r times r times r.
Careful: it is not 3 times r.
π is a special number for circles.
It is about 3.14.
So the volume is V = 4/3 × π × r³.
V = 43 × π × r3
Now put the numbers in: r = 3.
V = 43 × π × 33
3³ means 3 times 3 times 3.
That is 27.
33 = 3 × 3 × 3 = 27
So V = 4/3 × π × 27.
V = 43 × π × 27
Keep the numbers small: divide 27 by 3 first.
27 divided by 3 is 9.
273 = 9
So V = 4 × π × 9.
V = 4 × π × 9
4 times 9 is 36.
So V = 36π.
We keep π in the answer.
That is the exact answer.
V = 36π
2 About how much
To see about how much, put 3.14 for π.
V ≈ 36 × 3.14
36 times 3.14 is 113.04.
So the volume is about 113.04 cubic units.
V ≈ 113.04
Let's look back.
First: use the rule.
Last: about how much.
▶ Watch it solved
The question
Question: find the volume of a sphere (a ball) with diameter 6.
1 Find the radius
The diameter goes all the way across the circle.
The radius is half of it.
Half of 6 is 6 divided by 2.
r = 62
6 divided by 2 is 3.
So r = 3.
r = 3
2 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.
Worked out, that gives the rule for a ball.
The rule has r³: r times r times r.
Careful: it is not 3 times r.
π is a special number for circles.
It is about 3.14.
So the volume is V = 4/3 × π × r³.
V = 43 × π × r3
Now put the numbers in: r = 3.
V = 43 × π × 33
3³ means 3 times 3 times 3.
That is 27.
33 = 3 × 3 × 3 = 27
So V = 4/3 × π × 27.
V = 43 × π × 27
Keep the numbers small: divide 27 by 3 first.
27 divided by 3 is 9.
273 = 9
So V = 4 × π × 9.
V = 4 × π × 9
4 times 9 is 36.
So V = 36π.
We keep π in the answer.
That is the exact answer.
V = 36π
3 About how much
To see about how much, put 3.14 for π.
V ≈ 36 × 3.14
36 times 3.14 is 113.04.
So the volume is about 113.04 cubic units.
V ≈ 113.04
Let's look back.
First: find the radius.
Then: use the rule.
Last: about how much.
▶ Watch it solved
The question
Question: find the volume of a hemisphere (half a ball) with radius 3.
1 Half of a ball
A hemisphere is half of a ball.
A whole ball is 4/3 × π × r³.
π is a special number for circles.
It is about 3.14.
So the hemisphere is half of it: V = ½ × 4/3 × π × r³.
V = 12 × 43 × π × r3
Half of 4 thirds is 2 thirds.
43 ÷ 2 = 23
So V = 2/3 × π × r³.
V = 23 × π × r3
Now put the numbers in: r = 3.
V = 23 × π × 33
3³ means 3 times 3 times 3.
That is 27.
33 = 3 × 3 × 3 = 27
So V = 2/3 × π × 27.
V = 23 × π × 27
Keep the numbers small: divide 27 by 3 first.
27 divided by 3 is 9.
273 = 9
So V = 2 × π × 9.
V = 2 × π × 9
2 times 9 is 18.
So V = 18π.
We keep π in the answer.
That is the exact answer.
V = 18π
2 About how much
To see about how much, put 3.14 for π.
V ≈ 18 × 3.14
18 times 3.14 is 56.52.
So the volume is about 56.52 cubic units.
V ≈ 56.52
Let's look back.
First: half of a ball.
Last: about how much.
▶ Watch it solved
