Step by step
The question
Question: find the volume of a cone with radius 6 cm and slant height 10 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 right triangle.
So the Pythagorean theorem works: 6² + h² = 10².
6² means 6 times 6.
That is 36.
And 10² is 100.
Write the squares: 36 + h² = 100.
We want h² by itself.
The 36 is in the way.
To remove 36, we take away 36.
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.
36 and − 36 make zero.
Cross them out.
100 minus 36 is 64.
h² is 64.
So h is the square root of 64.
The square root of 64 is the positive number that times itself makes 64.
8 times 8 is 64.
So the height is h = 8 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.
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 = 6 and h = 8.
6² is 36, as before.
So V = ⅓ × π × 36 × 8.
Keep the numbers small: divide 36 by 3 first.
36 divided by 3 is 12.
So V = π × 12 × 8.
12 times 8 is 96.
So V = 96π.
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.
96 times 3.14 is 301.44.
So the volume is about 301.44 cm³.
Let's look back.
First: find the height.
Then: a third of the cylinder.
Last: about how much.