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edited Mar 18 in ~ 20:50

Andrei

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asked Mar 18 in ~ 20:40

JonasHausJonasHaus

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Given the edge $ heta$, split by the diameter comprise $B$, take into consideration the complying with diagram:

$overlineBO$ is the line through the center and $overlineBA$ is the chord cutting off the lune who area us wish to compute.

The area of the one wedge subtended by $angle BOA$ is$$fracpi- heta2r^2 ag1$$The area of $ riangle BOA$ is$$frac12cdotoverbracersinleft(frac heta2 ight)^ extaltitudecdotoverbrace2rcosleft(frac heta2 ight)^ extbase=fracsin( heta)2r^2 ag2$$Therefore, the area the the lune is $(1)$ minus $(2)$:$$fracpi- heta-sin( heta)2r^2 ag3$$To obtain the area separated into thirds, us want$$fracpi- heta-sin( heta)2r^2=fracpi3r^2 ag4$$which method we desire to solve$$ heta+sin( heta)=fracpi3 ag5$$whose solution deserve to be achieved numerically (e.g. Use $M=fracpi3$ and also $varepsilon=-1$ in this answer)$$ heta=0.5362669789888906 ag6$$Giving us

**Numerical Details**