SOLUTION: A chord at a distance 6 from the center of a circle with diameter 24 forms a segment (shaded) with its minor arc. What is the area of this segment?

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Question 328749: A chord at a distance 6 from the center of a circle with diameter 24 forms a segment (shaded) with its minor arc. What is the area of this segment?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!

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The area of the entire pie slice is equal to the area of the two triangle plus the remaining portion.
I'm not sure which area you require.
You can calculate the entire pie slice area by finding the angle C.
You can calculate the two triangles area by finding b.
Let's find b first.
From the diagram above,
6%5E2%2Bb%5E2=12%5E2
b%5E2=144-36
b=sqrt%28108%29
b=6sqrt%283%29
So then the area of the triangle is,
At=%281%2F2%29%286%29%286sqrt%283%29%29=18sqrt%283%29
You can calculate the angle C using trigonometry.
cos%28C%29=6%2F12=0.5
C=pi%2F3
For the entire circle, the area is Ac=pi%2812%29%5E2
Since we only have a 2C slice out of 2pi, then the area of the slice is
As=%28%282pi%2F3%29%2F%282pi%29%29%2Api%2A12%5E2
As=%28144%2F3%29pi
As=48pi
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So now we have all of the pieces.
As=2At%2BAr
where Ar is the area of the remaining portion.
48pi=2%2818sqrt%283%29%29%2BAr
Ar=48pi-36sqrt%283%29