Question

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

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.

Therefore 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.

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

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

⇒` "Area of circle"/"Area of equilateral triangle"=(pi(d/2)^2)/sqrt(3/4 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"=(pi xx a^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`