The height of a section of overlapping circles.

Question

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 $1/6$th 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!

Answer 1

If we draw a straight line between the two points of intersection, we have two circular segments joined at the bases.

The height $h$ of one of the circular segments is

$$h = R(1 - \cos (\theta/2))$$

where $\theta$ is the central angle and $R$ is the radius of the circle.

Can you take it from here?

Spoiler:

The central angle subtending the blue arc is the same as the angular measure of the arc, which is $\pi/3$. Taking $R=1$, the height of one of the circular segments is $h = 1 - \cos(\pi/6),$ or $h = 1 - (\sqrt{3}/2).$ Then, $y=2-\sqrt{3}.$