Here's the question you clicked on:
ConfusedFreshy
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!
|dw:1344217058690:dw|
Is the entire circle shaded?
No the right hand corner arc is not. That is kind of hard to see but it is not shaded up there.
Got it. And what's the 90 degrees?
I believe it is the central angle.
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 :/
|dw:1344217343489:dw|
Definitely, though the 90 degrees still seems somewhat unrelated to anything in the drawing…
I think the central angle is supposed to be 90 degrees. That is why it is there.
Meaning the angle formed by the two endpoints of the chord and the center, yes?
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?
Yes I know that.
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?
4, since the radius is 4.
Precisely. Can you get the area of that sector of the circle, ignoring the chord?
What I got could not be correct.
I got 720. Seeing that the formula for area of a sector is Area= [angle]radius^2/ 2
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?
Yes. Which would equal 180?
Well, area is pi * r^2, which here is pi * 4^2, which is pi * 16 right?
Yes. That divided by 4 is 4.
Right, so we have pi * 4, which is ~12.566
What does that mean? Is that the area of the whole sector?
Correct. So now let's figure out the area of the *unshaded* portion of the sector.
We can do that by finding the area of the triangle formed by the central angle and the chord, right?
Ummm. Pythagorean theorem so it could be 32= c^2
Well, that's how you'd find the length of the chord. We're just interested in the area of this: |dw:1344218326720:dw|
Omg. I just derped so hard there. The base and height of the triangle is 4 and 4. so it would be 8?
Exactly! So, how do we get the area of the unshaded area of that sector?
12.566 minus 8. 4.566 is the unshaded area?
Yep. And now the area of the shaded part of the circle?
You just lost me again. Which part am I looking for? THe rest of the circle?
Yeah, the shaded area of the circle is everything except the unshaded part we just found the area of right?
I seriously do not know. I feel so stupid whenever I do geometry. Is it 8?
Well, it's going to be pi*r^2 of the circle - the area we just calculated. Do you see how that works?
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.
Sounds about right!
Thank you so much!!
No problem! Hope that helped the reasoning for other problems!