Two Triangles in a Dodecagon

The difference between the orange and yellow areas is 2. What’s the total area of the regular dodecagon?

Let S be the area of the regular decagon.

The difference between the two areas is equal to the difference between the area of △ABC and the area of △ABD:

S(BCM) - S(AMD) = S(ABC) - S(ABD)

Since the hexagon is regular, we have that △ABD and △ABC are right-angle, and also ∠ABD = 15° and ∠ABC = ∠CAB = 45°.

If AB = 2R, then AD = 2\sqrt{2+\sqrt{3}}R, BD = 2\sqrt{2-\sqrt{3}}R, AC = BC = 2\sqrt{2}R. Therefore, S(ABC) = 4R^2 and S(ABD) = 2R^2. If S(ABC) - S(ABD) = 2, then 2R \times R = 2, and R = 1. Finally, S = 12S(AOD) = 6S(ABD) = 12.