Perimeter of a circular segment bounded by a square's side

Text

In a circle of radius $r=8$ cm consider the chord equal to the side of the inscribed square. Find the length of the boundary of the circular segment bounded by that chord and its (minor) arc.

Solution

The side of the inscribed square subtends a central angle of $90^\circ$; hence the chord is $\overline{AB}=r\sqrt2=8\sqrt2,$ and the $90^\circ$ arc has length $L=r\cdot\frac{\pi}{2}=8\cdot\frac{\pi}{2}=4\pi.$ The boundary is the sum: $p=8\sqrt2+4\pi\approx11.31+12.57=23.88\ \text{cm}.$ $\boxed{p=8\sqrt2+4\pi\approx23.88\ \text{cm}}$