What is the ratio of the areas of the incircle and circumcircle of a square?

$1: 2$

Incircle area square : Circumcircle area of square

In case of Incircle square, the diameter of circle will be side of square.

Thus Area of Circle A1 = Π * Side^2

In case of Circumcircle square, the diameter of circle will be diagonal of square.

$Diagonal^2 = Side^2 + Side^2$

Diagonal = sqrt(2*Side^2) = sqrt(2) * Side

Thus area of circle A2 = Π  * diagonal^2 = Π  * (sqrt(2) * Side)^2 = 2 Π Side^2

A1:A2 = Π Side^2 : 2Π Side^2 = 1:2

The correct answer is Option (4) → $1: 2$