Question

A cone with a radius of 8 feet and a slant height of 17 feet is shown below. What is the surface area of this cone? Use 22 over 7 for pi.

Answer 1

The area of a circle is \(\pi r^2\), but I think you know that.
So you can caclulate the area of the base circle, because you know its radius.

What about the "mantle"? (not sure how this is called, but you know what I mean...)
If you cut it open and spread it out, it will be part of a circle, with radius 17.
So calculating the area of a circle with radius 17 is easy as well.

The only problem is: what part of a circle is it?
It fits exactly on the base circle, so its circumference is equal to that of the base circle.
The circumference of a circle is \(2\pi r\), so that is \(16\pi\).
The circumference of a circle with radius 17 is therefore \(34\pi\).

This means that the "mantle" is only \(\dfrac{16\pi}{34\pi}=\dfrac{8}{17}th\) of a circle.

So its area is also 8/17 th of the area of a circle with radius 17.