Question

The area of a circle is the same as the area of a square. Their perimeters are in the ratio ______.

Options

• 1 : 1

• 2 : π

• π : 2

• `sqrt(pi) : 2`

Answer 1

The area of a circle is the same as the area of a square. Their perimeters are in the ratio `underlinebb(sqrt(pi) : 2)`.

Explanation:

Let a be the side of the square.

We know:
Area of a square = a² 

Let r be the radius of the circle.

We know:

Area of a circle`=pi"r"^2`

Because the area of the square is the same as the area of the circle, we have:

a² = πr²

`=> "r"^2/"a"^2 = 1/pi`

`=>"r"/"a" = 1/sqrt(pi)`

∴ Ratio of their perimeters

`=(2pi"r")/(4"a") ["Because perimeter of the circle is 2πr and permeter"  "of the square is 4"a"]`

`=(pi"r")/(2"a")`

`=pi/2xx"r"/"a"`

`=pi/2xx1/sqrt(pi)      ["Since" "r"/"a" = 1/sqrt(pi)]`

`=sqrt(pi)/2`

`= sqrt(pi) : 2`