Angle in figure consisting of a square surrounded by semi circles I'd like to know how to get the angle in the following problem:

It is a square with side equal to 1.  The radius of each semi circle is equal to the side of the square.  How can this angle be determined?

It's $30^\circ$. Let the point of intersection of "upper" arcs $BD$ and $AC$ be called $E$, and of upper $BD$ with lower arc $AC$ be called $F$. You should recognize that $\triangle ABE$ is equilateral (why?). What about $\triangle ADF$? Now finish.
