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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=`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`
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