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I chose to make $2$ segments of the unit, each of area a
quarter of the unit circle, and the remained part of the circle to
be divided into two equal parts, as shown in the figure.
Let $x$ be the central angle determined by the segment. The
area of the segment is: $${1\over 2}\left(x - \sin x\right).$$ And
it must be a quarter the area of the circle so: $${1\over 2}\left(x
- \sin x\right)= {\pi \over 4} .$$
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To find a better approximation I used this graph.
Plot $\left[x - \sin [x]- {\pi \over 2}, {x, 2.309,
2.31}\right]$
So $x \approx 2.3099$.
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