# What is the area of the Pokémon Moon logo?

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.

• Can you use the ruler to measure curves, e.g. to measure the diameter of the circle? Can you make extra marks on the diagram and measure to/from those? – 2012rcampion May 28 '16 at 16:03
• What do you mean by "measure curves"? You can't use it to measure the length of a curve, if that's what you're asking. – Joe Z. May 28 '16 at 16:03
• @kamenf The latter. The measurements you get with the straightedge can be used as infinitely precise for the theory of this problem. – Joe Z. May 28 '16 at 16:27
• @2012rcampion Yes, you can mark three arbitrary points on a circle and measure their distances. But you're not allowed a construct a point "whose distance is _________ from another point". – Joe Z. May 28 '16 at 16:45
• Should the question not be on the math overflowing website? – Adit Kirtani May 28 '16 at 18:33

In short:

I assume we can do calculations. So, we can calculate radius of any circle by inscribing arbitrary triangle in it. Let say it is $\triangle ABC$ with sides $a$, $b$ and $c$ which we can measure, and angles $\alpha,\;\beta$ and $\gamma$ which we may calculate form the sides:

$$\cos(\alpha)\;=\frac{a^2+b^2-c^2}{2ab}$$

$$R\;=\frac a{2\sin(\alpha)}$$