Radius of a circle (partially) inscribed off-centre in a square Suppose we have a square diamond. There is a circle, where the top point of the circle is at the top corner of the diamond, and it touches the bottom two sides of the diamond. What is the radius of the circle? !square and circle

I'm not good at posting diagrams so you will have to draw some additions to your own diagram. Draw the diagonal from top to bottom of the square; mark on it the centre of the circle; draw radii to the two points where the square is tangent to the circle. If the square has side $s$ and the circle has radius $r$, then finding the length of the diagonal in two ways gives $$s\sqrt2=r(1+\sqrt2)\ .$$ From this you can find the radius in terms of the side length.