What is the maximum area of an equilateral triangle that can be inscribed in a circle of radius 'a' ?

Question

What is the maximum area of an equilateral triangle that can be inscribed in a circle of radius $' a'$ ?

Smart Achievers · Accepted Answer

(3)

Ans. $\frac{3 \sqrt{3} a^{2}}{4}$

Explanation:

$Area of an equilateral triangle = \frac{\sqrt{3}}{4} \times {(Side)}^{2}$

$∴$ $Radius,$ $O A = O B = O C = a$

$Here,$ $O D = \frac{a}{2}$

As $O$ is the centroid of $∆ A B C$ ,

$∴$ $B D = \sqrt{O B^{2} - O D^{2}}$

$= \sqrt{a^{2} - \frac{a^{2}}{4}} = \frac{\sqrt{3} a}{2}$

Now, $B C = 2 B D = \sqrt{3} a$

So, $Area = \frac{\sqrt{3}}{4} \times {(\sqrt{3} a)}^{2}$

$= \frac{3 \sqrt{3}}{4} a^{2}$

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