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June 3rd, 2019, 12:31 AM  #1 
Newbie Joined: May 2019 From: Australia Posts: 8 Thanks: 0  The height of a section of overlapping circles.
Say I have two identical circles, both of radii of one, overlapping, as shown in the diagram below: In this diagram, x is the circumference of the circles, and the bit of the bottom circle which is drawn blue (the overlapping bit) is 1/6th of the whole circumference. What I'm looking for is y, which is this: Now, working out x is easy  it's 2 \pi r, thus the overlapping bit is 1/3 \pi r. But how do I proceed in finding y from here? Help is much appreciated! Thanks! 
June 3rd, 2019, 03:48 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,943 Thanks: 1132 Math Focus: Elementary mathematics and beyond 
Imagine a right triangle from the centre of a circle, the midpoint of the segment y and one of the points of intersection of the circles. This triangle has hypotenuse 1 and side opposite the centre of the circle 1/2. Also, this triangle is 306090. Can you use some basic trig and the Pythagorean theorem to find y?

June 3rd, 2019, 04:11 AM  #3  
Newbie Joined: May 2019 From: Australia Posts: 8 Thanks: 0  Quote:
 
June 3rd, 2019, 11:00 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 20,746 Thanks: 2133 
Can you post a diagram with the suggested triangle shown? Certain lines are radii, so they have length 1.

June 4th, 2019, 08:40 AM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,943 Thanks: 1132 Math Focus: Elementary mathematics and beyond 
$\displaystyle \frac{y}{2}=1\cos30=1\frac{\sqrt3}{2}=\frac{2\sqrt3}{2}\implies y=2\sqrt3$


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circles, height, overlapping, section 
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