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edited Mar 18 in ~ 20:50

Andrei

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Given the edge $\theta$, separation by the diameter include $B$, think about the adhering to diagram:

$\overlineBO$ is the line through the center and $\overlineBA$ is the chord cutting turn off the lune whose area we wish to compute.

The area the the circular wedge subtended by $\angle BOA$ is$$\frac\pi-\theta2r^2\tag1$$The area of $\triangle BOA$ is$$\frac12\cdot\overbracer\sin\left(\frac\theta2\right)^\textaltitude\cdot\overbrace2r\cos\left(\frac\theta2\right)^\textbase=\frac\sin(\theta)2r^2\tag2$$Therefore, the area the the lune is $(1)$ minus $(2)$:$$\frac\pi-\theta-\sin(\theta)2r^2\tag3$$To gain the area divided into thirds, we want$$\frac\pi-\theta-\sin(\theta)2r^2=\frac\pi3r^2\tag4$$which method we desire to solve$$\theta+\sin(\theta)=\frac\pi3\tag5$$whose solution have the right to be completed numerically (e.g. Use $M=\frac\pi3$ and also $\varepsilon=-1$ in this answer)$$\theta=0.5362669789888906\tag6$$Giving us

**Numerical Details**