- anonymous

Find the area of the shaded region. Point O marks the center of the circle. Picture below!
I need someone to give me the equation to follow to solve the problem please!

- chestercat

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- anonymous

|dw:1344217058690:dw|

- shadowfiend

Is the entire circle shaded?

- anonymous

No the right hand corner arc is not. That is kind of hard to see but it is not shaded up there.

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- shadowfiend

Got it. And what's the 90 degrees?

- anonymous

I believe it is the central angle.

- shadowfiend

Hm. I feel like I still don't have a good grasp of what the image looks like. Is there any chance you could either take a picture of it and upload it or draw it in a tool that can do the shading a little better and upload that image? We know our drawing tool needs a bit of work :/

- anonymous

|dw:1344217343489:dw|

- anonymous

Is that better?

- shadowfiend

Definitely, though the 90 degrees still seems somewhat unrelated to anything in the drawingâ€¦

- anonymous

I think the central angle is supposed to be 90 degrees. That is why it is there.

- shadowfiend

Meaning the angle formed by the two endpoints of the chord and the center, yes?

- anonymous

Yes.

- shadowfiend

Ok, then I think what you can do is, because it's a 90 degree angle you know that the circle portion in question is 1/4 the circle, right?

- anonymous

Yes I know that.

- shadowfiend

And in order for that to be true, what does the length of the line segment from the center of the circle to the endpoint of the chord have to be?

- anonymous

4, since the radius is 4.

- shadowfiend

Precisely. Can you get the area of that sector of the circle, ignoring the chord?

- anonymous

What I got could not be correct.

- shadowfiend

What did you get?

- anonymous

I got 720.
Seeing that the formula for area of a sector is
Area= [angle]radius^2/ 2

- shadowfiend

I'll leave aside whether that formula is correct or not, in this particular case we've already determined that this is a quarter of the circle, so really it's just area / 4, right?

- anonymous

Yes. Which would equal 180?

- shadowfiend

Well, area is pi * r^2, which here is pi * 4^2, which is pi * 16 right?

- anonymous

Yes.
That divided by 4 is 4.

- shadowfiend

Right, so we have pi * 4, which is ~12.566

- anonymous

What does that mean? Is that the area of the whole sector?

- shadowfiend

Correct. So now let's figure out the area of the *unshaded* portion of the sector.

- shadowfiend

We can do that by finding the area of the triangle formed by the central angle and the chord, right?

- anonymous

Ummm. Pythagorean theorem so it could be 32= c^2

- shadowfiend

Well, that's how you'd find the length of the chord. We're just interested in the area of this:
|dw:1344218326720:dw|

- anonymous

Omg. I just derped so hard there.
The base and height of the triangle is 4 and 4. so it would be 8?

- shadowfiend

Exactly! So, how do we get the area of the unshaded area of that sector?

- anonymous

12.566 minus 8.
4.566 is the unshaded area?

- shadowfiend

Yep. And now the area of the shaded part of the circle?

- anonymous

You just lost me again. Which part am I looking for? THe rest of the circle?

- shadowfiend

Yeah, the shaded area of the circle is everything except the unshaded part we just found the area of right?

- anonymous

I seriously do not know. I feel so stupid whenever I do geometry.
Is it 8?

- shadowfiend

Well, it's going to be pi*r^2 of the circle - the area we just calculated. Do you see how that works?

- shadowfiend

Basically, to find the area of the circle except for that portion, we worked backwards to find the area of that portion, then subtract it from the area of the circle as a whole.

- anonymous

so its 45.699?

- shadowfiend

Sounds about right!

- anonymous

Thank you so much!!

- shadowfiend

No problem! Hope that helped the reasoning for other problems!

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