Perimeter of congruent region

For the congruent region, the longer arc is one sixth of the circumference, which is $\frac{\pi}{3}$ , while the shorter arc is half of the longer arc. Hence its length is $\frac{\pi}{6}$ .

Note that the segment of the congruent region is the difference of chord and radius of the unit circle. For the chord, it’s length is double of the height of the equilateral triangle of unit side length. Hence the chord is of length $\sqrt{3}$ .

Now the perimeter of the congruent region is $\frac{\pi}{3}+\frac{\pi}{6}+\left(\sqrt{3}-1\right) \approx 2.303$ .

Watch this video for this explanation and discussion.