Step by step
The question
Question: find the volume of a cone with radius 5 cm and slant height 13 cm.
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 triangle.
The height stands straight up from the middle of the base, so it has a square corner.
Pythagoras: in a right triangle, square the two sides at the square corner and add them.
That makes the long side squared.
The long side is the slant height, 13.
So 5² + h² = 13².
5² means 5 times 5.
That is 25.
And 13² is 169.
Write the squares: 25 + h² = 169.
Is h alone now?
Not yet: + 25 is still with h.
How do we remove it?
We want h² by itself.
The 25 is in the way.
To remove 25, we take away 25.
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.
25 and − 25 cancel each other.
They make zero.
169 minus 25 is 144.
h² is 144.
So h is the square root of 144.
The square root of 144 is the plus number that times itself makes 144.
12 times 12 is 144.
−12 times −12 is 144 too, but the √ sign always means the plus one.
So the height is h = 12 cm.
So after step 1: we took away 25, and now we have h = 12 cm.
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: the way round is about 3.14 times the way across.
The cylinder is circle area times height: π × r² × h.
So the cone is V = ⅓ × π × r² × h.
Now put the numbers in: r = 5 and h = 12.
5² is 25, as before.
So V = ⅓ × π × 25 × 12.
⅓ of 12 means 12 divided by 3.
We can multiply in any order.
So we do that first: it keeps the numbers small.
12 divided by 3 is 4.
So ⅓ × 12 becomes 4.
So V = π × 25 × 4.
25 times 4 is 100.
So V = 100π.
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.
100 times 3.14 is 314.
So the volume is about 314 cm³.
Let's look back.
First: find the height.
Then: a third of the cylinder.
Last: about how much.