Step by step
The question
Question: find the surface area of a cone with radius 5 and height 12.
1 Find the slant height
First we need the slant height: down the side.
We call it l.
The radius, the height and the slant height make a right triangle.
So the Pythagorean theorem works: 5² + 12² = l².
5² means 5 times 5.
That is 25.
And 12² is 144.
Write the squares: 25 + 144 = l².
25 plus 144 is 169.
l² is 169.
So l is the square root of 169.
The square root of 169 is the positive number that times itself makes 169.
13 times 13 is 169.
So the slant height is l = 13.
2 Count the parts
The surface area is the area of all the outside.
A cone has a circle at the bottom and a curved side.
3 The base
The base is a circle with radius 5.
A circle is π × r².
Now put the numbers in: r = 5.
5² means 5 times 5.
That is 25.
So the base is 25π.
4 The curved side
Cut the curved side open and lay it flat.
It is part of a bigger circle.
Its area is π × r × l.
Careful: l is the slant height, down the side.
Not the height inside the cone.
π is a special number for circles.
It is about 3.14.
So the curved side is π × r × l.
Now put the numbers in: r = 5 and l = 13.
5 times 13 is 65.
So the side is 65π.
5 Add the parts
Now we add the base and the curved side.
25 plus 65 is 90.
So S = 90π.
We keep π in the answer.
That is the exact answer.
6 About how much
To see about how much, put 3.14 for π.
90 times 3.14 is 282.6.
So the surface area is about 282.6 square units.
Let's look back.
First: find the slant height.
Then: count the parts.
Then: the base.
Then: the curved side.
Then: add the parts.
Last: about how much.