Step by step
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 means 3 times 3.
That is 9.
And 5² is 25.
Write the squares: 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 and − 9 make zero.
Cross them out.
25 minus 9 is 16.
h² is 16.
So h is the square root of 16.
The square root of 16 is the positive number that times itself makes 16.
4 times 4 is 16.
So the height is 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.
Now put the numbers in: r = 3 and h = 4.
3² is 9, as before.
So V = ⅓ × π × 9 × 4.
Keep the numbers small: divide 9 by 3 first.
9 divided by 3 is 3.
So V = π × 3 × 4.
3 times 4 is 12.
So V = 12π.
We keep π in the answer.
That is the exact answer.
3 About how much
To see about how much, put 3.14 for π.
12 times 3.14 is 37.68.
So the volume is about 37.68 cubic units.
Let's look back.
First: find the height.
Then: a third of the cylinder.
Last: about how much.