Venn diagram

Shape and space, angle and circle properties, ...
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carl256
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Joined: Sun May 06, 2012 6:12 am

Venn diagram

Post by carl256 » Fri Jan 18, 2013 12:48 pm

Imagine two congruent circles that intersect at exactly two points A and B. One of the circles has a center C. What is the measure of angle ACB such that the area of the intersection between the two circles is equal to the area of one circle minus the area of intersection?
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TheEvil
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Location: Szeged, Hungary

Re: Venn diagram

Post by TheEvil » Fri Jan 18, 2013 6:51 pm

If you call that angle [alpha] and measure it in radian, then it is the unique solution to:
[alpha] - sin ([alpha]) =[frac]pi,2[/frac]
Wolfram Alpha give: [alpha]=2.30988 which is about 132.3464 degrees.
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carl256
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Re: Venn diagram

Post by carl256 » Sat Jan 19, 2013 2:00 am

Can you tell me how you worked out that equation? And what 2.30988 is in terms of pi?

Thanks a lot!
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TheEvil
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Location: Szeged, Hungary

Re: Venn diagram

Post by TheEvil » Sat Jan 19, 2013 9:34 am

Let us fix that every angle is measured in radians. If the intersection of the two circles halves the circles then the area between the chord AB and the arc (which is exactly the half of the intersection), is a quarter of the area of the circle. Let us call that area T, and the common radius r. We have
4T=r2[pi].
On the other hand, the area of T (which is well-known, see also: http://en.wikipedia.org/wiki/Circular_segment):
T=[frac]1,2[/frac]r2([alpha]-sin([alpha])).
From these two equalities, we get the desired one (which is solved by Wolfram Alpha).

To turn 2.30988 into the terms of pi, the routine works:
2.30988=[frac]2.30988,[pi][/frac]*[pi]=0.73526*[pi].

Hope you could understand that, but I am not a native speaker.
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