Step by step
The question
Question: find the volume of a cone with radius 8 and slant height 10.
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: 8² + h² = 10².
8² means 8 times 8.
That is 64.
And 10² is 100.
Write the squares: 64 + h² = 100.
We want h² by itself.
The 64 is in the way.
To remove 64, we take away 64.
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.
64 and − 64 make zero.
Cross them out.
100 minus 64 is 36.
h² is 36.
So h is the square root of 36.
The square root of 36 is the positive number that times itself makes 36.
6 times 6 is 36.
So the height is h = 6.
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 = 8 and h = 6.
8² is 64, as before.
So V = ⅓ × π × 64 × 6.
Keep the numbers small: divide 6 by 3 first.
6 divided by 3 is 2.
So V = π × 64 × 2.
64 times 2 is 128.
So V = 128π.
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.
128 times 3.14 is 401.92.
So the volume is about 401.92 cubic units.
Let's look back.
First: find the height.
Then: a third of the cylinder.
Last: about how much.