Question 14

What is the area of an equilateral triangle whose inscribed circle has radius R?

Solution

If a is the length of equilateral triangle, height is $$\frac{\sqrt{3}a}{2}$$

Inradius = $$\frac{\sqrt{\ 3}a}{2}\left(\frac{1}{3}\right)=\frac{a}{2\sqrt{3}}$$

It is given, R = $$\frac{a}{2\sqrt{3}}$$

Area of triangle = Inradius * semiperimeter = $$R\left(\frac{3a}{2}\right)$$ = $$\frac{3R}{2}\left(2\sqrt{3}R\right)=3\sqrt{3}R^2$$

The answer is option D.

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