Worked examples
The question
Question: find the volume of a cone with radius 3 and height 4.
1 A third of the cylinder
Draw a cylinder around the cone, with the same base and the same height.
Fill the cone with water 3 times.
That fills the cylinder exactly.
So the cone is one third of the cylinder.
π is a special number for circles.
It is about 3.14.
The cylinder is π × r² × h.
So the cone is V = ⅓ × π × r² × h.
V = 13 × π × r2 × h
Now put the numbers in: r = 3 and h = 4.
V = 13 × π × 32 × 4
3² means 3 times 3.
That is 9.
32 = 3 × 3 = 9
So V = ⅓ × π × 9 × 4.
V = 13 × π × 9 × 4
Keep the numbers small: divide 9 by 3 first.
9 divided by 3 is 3.
93 = 3
So V = π × 3 × 4.
V = π × 3 × 4
3 times 4 is 12.
So V = 12π.
We keep π in the answer.
That is the exact answer.
V = 12π
2 About how much
To see about how much, put 3.14 for π.
V ≈ 12 × 3.14
12 times 3.14 is 37.68.
So the volume is about 37.68 cubic units.
V ≈ 37.68
Let's look back.
First: a third of the cylinder.
Last: about how much.
▶ Watch it solved
The question
Question: find the volume of a cone with diameter 6 and height 4.
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 A third of the cylinder
Draw a cylinder around the cone, with the same base and the same height.
Fill the cone with water 3 times.
That fills the cylinder exactly.
So the cone is one third of the cylinder.
π is a special number for circles.
It is about 3.14.
The cylinder is π × r² × h.
So the cone is V = ⅓ × π × r² × h.
V = 13 × π × r2 × h
Now put the numbers in: r = 3 and h = 4.
V = 13 × π × 32 × 4
3² means 3 times 3.
That is 9.
32 = 3 × 3 = 9
So V = ⅓ × π × 9 × 4.
V = 13 × π × 9 × 4
Keep the numbers small: divide 9 by 3 first.
9 divided by 3 is 3.
93 = 3
So V = π × 3 × 4.
V = π × 3 × 4
3 times 4 is 12.
So V = 12π.
We keep π in the answer.
That is the exact answer.
V = 12π
3 About how much
To see about how much, put 3.14 for π.
V ≈ 12 × 3.14
12 times 3.14 is 37.68.
So the volume is about 37.68 cubic units.
V ≈ 37.68
Let's look back.
First: find the radius.
Then: a third of the cylinder.
Last: about how much.
▶ Watch it solved
The question
Question: find the volume of a cone with radius 3 and slant height 5.
1 Find the height
First we need the height: straight up the middle.
We call it h.
The radius, the height and the slant height make a right triangle.
So the Pythagorean theorem works: 3² + h² = 5².
32 + h2 = 52
3² means 3 times 3.
That is 9.
And 5² is 25.
32 = 3 × 3 = 9
52 = 5 × 5 = 25
Write the squares: 9 + h² = 25.
9 + h2 = 25
We want h² by itself.
The 9 is in the way.
To remove 9, we take away 9.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
9 + h2 − 9 = 25 − 9
9 and − 9 make zero.
Cross them out.
25 minus 9 is 16.
h2 = 16
h² is 16.
So h is the square root of 16.
h = √16
The square root of 16 is the positive number that times itself makes 16.
4 times 4 is 16.
4 × 4 = 16
So the height is h = 4.
h = 4
2 A third of the cylinder
Draw a cylinder around the cone, with the same base and the same height.
Fill the cone with water 3 times.
That fills the cylinder exactly.
So the cone is one third of the cylinder.
π is a special number for circles.
It is about 3.14.
The cylinder is π × r² × h.
So the cone is V = ⅓ × π × r² × h.
V = 13 × π × r2 × h
Now put the numbers in: r = 3 and h = 4.
V = 13 × π × 32 × 4
3² is 9, as before.
So V = ⅓ × π × 9 × 4.
V = 13 × π × 9 × 4
Keep the numbers small: divide 9 by 3 first.
9 divided by 3 is 3.
93 = 3
So V = π × 3 × 4.
V = π × 3 × 4
3 times 4 is 12.
So V = 12π.
We keep π in the answer.
That is the exact answer.
V = 12π
3 About how much
To see about how much, put 3.14 for π.
V ≈ 12 × 3.14
12 times 3.14 is 37.68.
So the volume is about 37.68 cubic units.
V ≈ 37.68
Let's look back.
First: find the height.
Then: a third of the cylinder.
Last: about how much.
▶ Watch it solved
