A good question!
[1] Let's make the problem easier to start. Imagine that they had said to find the area of a sector bounded by a 360* arc (the whole circle). Could we have found the area then?
Sure we could! We know that A = pi*r^2, so that would just be:
A = pi*(6)^2 = 36*pi sq. miles.
Easy. But of course, we weren't so lucky... they want 135*... drat!
[2] Well, let's get a little closer, could we find the area if the angle was 180*? Sure! That's just half the circle. After all:
180*/360* = 1/2.
We know the whole circle is 36*pi sq. miles. So, the area bounded by a 180* arc would just be half of this: 18*pi sq. miles.
"But that's not the question," you scream!! Alright, alright, calm down... let's bring it all together.
[3] We know the area of the whole circle [1]. We also know that if we can figure out the fraction of the circle the problem is easy [2]. So, what fraction of a circle would match an arc of 135*?
Well, we can see that
135*/360* = 0.375 or 3/8
So the area is just 0.375 of the whole circle:
0.375* 36*pi sq. miles = 13.5*pi sq. miles
Finally... we're done!
