You are watching: How to divide a circle into 3 equal parts

geometry circles area angle

Share

Cite

Follow

edited Mar 18 at 20:50

Andrei

28.5k44 gold badges2121 silver badges4444 bronze badges

asked Mar 18 at 20:40

JonasHausJonasHaus

21

$\endgroup$

1

Add a comment |

## 4 Answers 4

Active Oldest Votes

1

See more: How Would You Describe The Cunninghams In To Kill A Mockingbird York Notes

$\begingroup$

Given the angle $\theta$, split by the diameter containing $B$, consider the following diagram:

$\overline{BO}$ is the line through the center and $\overline{BA}$ is the chord cutting off the lune whose area we wish to compute.

The area of the circular wedge subtended by $\angle BOA$ is$$\frac{\pi-\theta}2r^2\tag1$$The area of $\triangle BOA$ is$$\frac12\cdot\overbrace{r\sin\left(\frac\theta2\right)}^\text{altitude}\cdot\overbrace{2r\cos\left(\frac\theta2\right)}^\text{base}=\frac{\sin(\theta)}2r^2\tag2$$Therefore, the area of the lune is $(1)$ minus $(2)$:$$\frac{\pi-\theta-\sin(\theta)}2r^2\tag3$$To get the area divided into thirds, we want$$\frac{\pi-\theta-\sin(\theta)}2r^2=\frac\pi3r^2\tag4$$which means we want to solve$$\theta+\sin(\theta)=\frac\pi3\tag5$$whose solution can be achieved numerically (e.g. use $M=\frac\pi3$ and $\varepsilon=-1$ in this answer)$$\theta=0.5362669789888906\tag6$$Giving us

**Numerical Details**