Question

If a square is inscribed in a circle, find the ratio of the areas of the circle and the square.

Answer 1

If a square is inscribed in a circle, then the diagonals of the square are diameters of the circle.

Let the diagonal of the square be d cm.

Thus, we have:

Radius , `"r" = "d"/2 "cm"` 

Area of the circle = πr² 

    `= pi"d"^2/4  "cm"^2`

We know ;  

`"d"= sqrt(2)xx"Side" `

`=> "Side" = ("d"/sqrt(2)) "cm"`

Area of the the circle = πr² 

`=pi"d"^2/4 "cm"^2`

We know ;

`d = sqrt(2)xx"side"`

`⇒ "Side" = "d"/sqrt(2) "cm"`

Area of the square`=("Side")^2`

`=("d"/sqrt(2))^2`

`= "d"^2/2 "cm"^2`

Ratio of the area of the circle to that of the square:

`= (pi"d"^2/4)/("d"^2/2)`

`= pi/2 `

Thus, the ratio of the area of the circle to that of the square is π : 2