Geometry question w/ a Circle. find area of a shaded region. (Harder problem)

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Geometry question w/ a Circle. find area of a shaded region. (Harder problem)

Mathematics
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|dw:1345779977023:dw|
I think what you have to do is to bisect the triangle into 60 degree up top and then the bottom left and right are still 30 degrees. then you do the 30 60 90 triangle thing.
i am stuck from there

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Other answers:

i did it a different way...
well explain.
with integrals...
well. it shouldn't be that complicated im only doing geometry.
Area of minor segment \[A = \frac{1}{2} r^2 \left(\theta-\sin\theta\right) \] But the only problem is \(r\) is not given..
I think I meant to post the area of a sector of a circle
you guys are thinking too hard its only like subtracting a sector
Area of sector - area of triangle
Yeah that
But \(r\) is not given!!! Im thinking of using the cosine rule and sine rule to find \(r\) would it work?
here ill take it a bit further. |dw:1345780578744:dw| does that help?
It will not be hard to find \(r\)
Is the vertex of angle 120 is the center of the circle?
yep
|dw:1345780720301:dw| looks like you can use the sine rule..
Okay, so yeah just do Area of Sector - Area of Triangle like what @Mimi_x3 suggested
soo all we need is the area of the circle and then we can use the 120/360 * pi r2 or something and subtract the area from that.
lets find \(r\) first
|dw:1345780842471:dw|
I got 8 as well using the sine rule
Now its easy..you can do it yourself
ok so 64 pi is the area of the circle
and the area of the triangle left side is 1/2 * 4sqroot3 * 4
*2 to make up for the other side.
I will just use this formula\[A = \frac{1}{2} r^2 \left(\theta-\sin\theta\right) \] No need to derive it
ok
and it must be in radians..so convert it..
oh heck nah jose
you guys are thinking too hard o.0
oh wait i got the answer. just do 120/360 which is 1/3 and then i did 1/3 * 64pi which is the area of the circle and it is 64pi/3 and now subtract it by the area of the triangle which is
1/2 4sqrt3 * 4 or 8sqrt3 * 2 for the other side
|dw:1345781211630:dw| |dw:1345781273195:dw|
|dw:1345781325118:dw|
so the final answer is \[64\pi/3 - 16\sqrt{3}\]
yes good job @Lethal
yep, i would best response both of u but i can only do one. thx alex.

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