The side of the square inscribed in a square be a units then the required ratio is π : 2.
A square is inscribed in a circle or a polygon if its four vertices are on the circle's circumference or the polygon's sides.
Inscribed in the circle is a square that fits snugly inside a circle. The square's corners will touch but not intersect the circle's boundary, and the diagonal of the square will equal the diameter of the circle. Furthermore, as with any square's diagonal, it equals the hypotenuse of a 45°-45°-90° triangle.
Let the side of the square inscribed in a square be a units.
Hence, the required ratio is 2 : 1.
Let ABCD be a square inscribed in a circle of radius 'r'.
Now, the diameter of the circle is the diagonal of the square.
Therefore, BD = 2r.
In △BDC, using Pythagoras theorem
[tex]BC^2+CD^2=BD^2[/tex]
⇒ [tex]a^2+a^2=(2r)^2[/tex]
⇒ [tex]2a^2=4r^2[/tex]
⇒ [tex]a^2=2r^2[/tex]
Area of square=[tex]2r^2[/tex]
Area of circle =[tex]$\pi r^2[/tex]
Required ration = [tex]$\pi r^2:2r^2[/tex] = π : 2
Therefore, the required ration is π : 2.
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