Step by step
The question
Question: find the surface area of a cone with radius 3 and height 4.
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: 3² + 4² = l².
3² means 3 times 3.
That is 9.
And 4² is 16.
Write the squares: 9 + 16 = l².
9 plus 16 is 25.
l² is 25.
So l is the square root of 25.
The square root of 25 is the positive number that times itself makes 25.
5 times 5 is 25.
So the slant height is l = 5.
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 3.
A circle is π × r².
Now put the numbers in: r = 3.
3² means 3 times 3.
That is 9.
So the base is 9π.
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 = 3 and l = 5.
3 times 5 is 15.
So the side is 15π.
5 Add the parts
Now we add the base and the curved side.
9 plus 15 is 24.
So S = 24π.
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.
24 times 3.14 is 75.36.
So the surface area is about 75.36 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.