Recently, the seventh generation of Pokémon games — Pokémon Sun and Moon — were announced. Along with it came the new logo symbols, of which the Moon logo is of particular interest:
Consider the symbol at the right, a high-resolution version of which is shown below (from here):
All the borders in this logo are created from circular arcs. Suppose the large circle that the logo was "carved out of" was radius 1; how would you figure out the exact area of this rounded figure using only a straightedge with ruler markings on it (that can measure any straight-line distance between two given points)? In particular, you're not given the radii or centers of curvature of any of the arcs.