Question

If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square?

 

Answer 1

We have the following situation 

Let BD be the diameter and diagonal of the circle and the square respectively.

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

Area of the square = `"side"^2`

As we know that diagonal of the square is the diameter of the square.

Diagonal=`2r`

Side of the square= `"diagonal"/sqrt2............(1)`

Substituting "diagonal"=2r in equation (1) we get,

side of the square=`(2r)/sqrt2`

Now we will find the ratio of the areas of circle and square. 

Area of circle/Area of square=`(pir^2)/((2r)/sqrt2)^2`

Now we will simplify the above equation as below,

Area of circle/Area of square=`(pir^2)/(r^2/2)`

Area of circle/Area of square=`pir^2xx2/(4r^2)`

Hence, `"Area of circle"/"Area of square"-pi/2` 

Therefore, ratio of areas of circle and square is `pi:2`