Verified Answer :

jtvatsim

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 = pir^2, so that would just be:
A = pi(6)^2 = 36pi 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 36pi sq. miles. So, the area bounded by a 180 arc would just be half of this: 18pi 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 36pi sq. miles = 13.5*pi sq. miles

Finally... we're done!

williamwu309

If the radius of the circle is 6 miles (Wow, that is a big circle!!!), then the formula to find the area is \(\pi r^2\), and that would give us the area of \(36 \pi\) square miles, and \(\frac{360}{135}\) equals \(\frac{8}{3}\), and \(\frac{36}{/frac{8}{3}}\) equals \(36 \cdot \frac{3}{8}}\) , and that equals \(\frac{108}{8}\), which would equal \(\frac{27}{2}\), or \(13.5\)

P.S. This problem is harder than other problems, but I hope this helped!!! ^^

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