The ratio of the areas of a circle and an equilateral

Solution:

We are given that diameter and side of an equilateral triangle are equal.

Let d and a are the diameter and side of circle and equilateral triangle respectively.

$\therefore d=a$

We know that area of the circle $=\pi r^{2}$

Area of the equilateral triangle $=\frac{\sqrt{3}}{4} a^{2}$

Now we will find the ratio of the areas of circle and equilateral triangle.

$\therefore \frac{\text { Area of circle }}{\text { Area of equilateral triangle }}=\frac{\pi r^{2}}{\frac{\sqrt{3}}{4} a^{2}}$

We know that radius is half of the diameter of the circle.

$\therefore \frac{\text { Area of circle }}{\text { Area of equilateral triangle }}=\frac{\pi\left(\frac{d}{2}\right)^{2}}{\frac{\sqrt{3}}{4} a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of equilateral triangle }}=\frac{\pi \times \frac{d^{2}}{4}}{\frac{\sqrt{3}}{4} a^{2}}$

Now we will substitute  in the above equation,

$\frac{\text { Area of circle }}{\text { Area of equilateral triangle }}=\frac{\pi \times \frac{a^{2}}{4}}{\frac{\sqrt{3}}{4} a^{2}}$

$\therefore \frac{\text { Area of circle }}{\text { Area of equilateral triangle }}=\frac{\pi}{\sqrt{3}}$

Therefore, ratio of the areas of circle and equilateral triangle is $\pi: \sqrt{3}$.

Hence, the correct answer is option (b).