 # What is the ratio of the areas of a circle and an Question:

What is the ratio of the areas of a circle and an equilateral triangle whose diameter and a side are respectively equal?

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.

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

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

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

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

Now we will substitute $d=a$ in the above equation,

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

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

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