Step by step
The question
Question: find the surface area of a cone with radius 8 m and height 6 m.
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: 8² + 6² = l².
8² means 8 times 8.
That is 64.
And 6² is 36.
Write the squares: 64 + 36 = l².
64 plus 36 is 100.
l² is 100.
So l is the square root of 100.
The square root of 100 is the positive number that times itself makes 100.
10 times 10 is 100.
So the slant height is l = 10 m.
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 8 m.
A circle is π × r².
Now put the numbers in: r = 8.
8² means 8 times 8.
That is 64.
So the base is 64π.
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 = 8 and l = 10.
8 times 10 is 80.
So the side is 80π.
5 Add the parts
Now we add the base and the curved side.
64 plus 80 is 144.
So S = 144π.
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.
144 times 3.14 is 452.16.
So the surface area is about 452.16 m².
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.