## Venn diagram

Shape and space, angle and circle properties, ...
carl256
Posts: 6
Joined: Sun May 06, 2012 7:12 am

### Venn diagram

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? TheEvil
Posts: 84
Joined: Sun Nov 13, 2011 10:38 am
Location: Szeged, Hungary

### Re: Venn diagram

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. carl256
Posts: 6
Joined: Sun May 06, 2012 7:12 am

### Re: Venn diagram

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

Thanks a lot! TheEvil
Posts: 84
Joined: Sun Nov 13, 2011 10:38 am
Location: Szeged, Hungary

### Re: Venn diagram

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. 