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Find the area of the polygon

Find the area of the polygon-example-1
User Cacovsky
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Looks like the larger triangle is equilateral (well, it must be, otherwise finding the area is impossible). That means the inradius is 6 and the outradius is 12, which is to say the altitudes (equal to the height, in this case) of this triangle have length 6 + 12 = 18.

In the smaller right triangle, the length of the missing leg is half the base of the larger triangle; call this length
\frac b2 By the Pythagorean theorem,


6^2+\left(\frac b2\right)^2=12^2\implies b=12\sqrt3

Then the area of the larger triangle is


\frac12(12\sqrt3)(18)=\boxed{108\sqrt3}

User Alasdair McLeay
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