Question

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

Options

• \[\pi: \sqrt{2}\] 

• \[\pi: \sqrt{3}\]

• \[\sqrt{3}: \pi\]

• \[\sqrt{2}: \pi\]

Answer 1

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. 

`∴ d=a` 

We know that area of the circle =`pir^2`

Area of the equilateral triangle =`sqrt3/4 a^2`

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

`∴ "Area of circle"/"Area of equilateral triangle"=(pir^2)/(sqrt3/4 a)`

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

`∴ "Area of circle"/"Area of equilateral triangle"=(pi (d/2)^2)/(sqrt3/4 a^2)`

`∴ "Area of circle"/"Area of equilateral triangle"=(pixxd^2/4)/(sqrt3/4 a^2)`

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

`∴ "Area of circle"/"Area of equilateral triangle"= ( pixxa^2/4)/(sqrt3/4 a^2)`

`∴ "Area of circle"/"Area of equilateral triangle"=pi/sqrt3`

Therefore, ratio of the areas of circle and equilateral triangle is  `pi:sqrt3`