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A circle is inscribed in an equilateral triangle, and another is circumscribed about the same triangle. If the length of the triangle's altitude is 6, determine the area of the region that is not occupied by the smaller circle.

 Feb 23, 2021
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The side length is $\frac{6}{\sqrt{3}}=2\sqrt{3}$. And the area is $6\sqrt{3}$ from there. The circumradius is then $\frac{2\sqrt{3}}{\sqrt{3}}=2$. The inradius is equal to the area divided by the semiperimeter, which is $1$. The result is $4\pi-\pi=\boxed{3\pi}$.

 Feb 23, 2021

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