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in Perimeter and Area of Plane Figures by (23.5k points)
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What is the ratio of the area of a circum circle of equilateral triangle to the area of the square with same side length as equilateral triangle?

(a) π : √3 

(b) √3 : 2 

(c) 1 : √2 

(d) π : 3

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(d)  π : 3.

Let each side of the equilateral Δ be a units. 

Then, circumradius of the circle = \(\frac{\text{side}}{\sqrt3}\) = \(\frac{a}{\sqrt3}\) units

∴ Area of circumcircle = \(\pi\bigg(\frac{a}{\sqrt3}\bigg)^2\) = \(\frac{\pi{a}^2}{3}\) sq units

Area of square of side a units = a2 sq units

∴ Required ratio = \(\frac{\frac{\pi{a}^2}{3}}{a^2}\) = \(\frac{\pi}{3}\) = π : 3.

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