Question : The ratio of the area of an equilateral triangle and that of its circumcircle is:
Option 1: $2\sqrt3:2\pi$
Option 2: $4:\pi$
Option 3: $3\sqrt3:4\pi$
Option 4: $7\sqrt2:2\pi$
Correct Answer: $3\sqrt3:4\pi$
Solution :
Let the side of the equilateral triangle be $a$ units and the radius of its circumcircle be $r$ units.
Now, the area of the equilateral triangle = $\frac{\sqrt3}{4}a^2$ sq.units
The radius of the circumcircle = $r=\frac{a}{\sqrt3}$ units
So, the area of the circumcircle = $\pi×(\frac{a}{\sqrt3})^2=\frac{\pi a^2}{3}$ units
$\therefore$ The required ratio = $\frac{\sqrt3}{4}a^2:\frac{\pi a^2}{3}=3\sqrt3:4\pi$
Hence, the correct answer is $3\sqrt3:4\pi$.
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