Question

The difference between the area of the circumscribed circle and the area of the inscribed circle of an equilateral triangle is 2156 sq. cm. What is the area of the equilateral triangle? Assume that $\pi = \frac{22}{7}$

Accepted Answer

A. For an equilateral triangle of side $a$, circumradius $R = \frac{a}{\sqrt{3}}$ and inradius $r = \frac{a}{2\sqrt{3}}$. Difference in area = $\pi R^2 - \pi r^2 = \pi \left(\frac{a^2}{3} - \frac{a^2}{12}ight) = \frac{\pi a^2}{4} = 2156$. $$\frac{22}{7} \cdot \frac{a^2}{4} = 2156 \implies a^2 = \frac{2156 \cdot 28}{22} = 2744$$ Area of equilateral triangle = $\frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} (2744) = 686\sqrt{3} 	ext{ sq. cm.}$$

Geometry: Equilateral Triangle Circles

Explanation

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