What is the ratio of the inradius and circumradius of a square?

$1:\sqrt{2}$

We know that,

Diagonal of the square = \(\sqrt {2}\)side

The inradius of the square = \(\frac{side}{ 2}\)

and the circumradius of the square = \(\frac{Diagonal}{2}\)

So, the ratio of the inradius and circumradius of a square = \(\frac{side}{ \sqrt {2} }\) : \(\frac{Diagonal}{2}\) 

= \(\frac{side}{ 2 }\) : \(\frac{\sqrt {2}side}{2}\)

= 1 : \(\sqrt {2}\)