Quick facts for circle inscribed in a square and vice versa.

I often spend way too much time re-solving for these, each time I encounter them.

Circle Inscribed In A Square
The circle will have radius r, and consequentially the square will have side 2r.
What this means, is:
Ratio of area of circle to area of square = (pi)r^2 to 4r^2, or (pi)/4
Ratio of perimeter of circle to perimeter of square = 2(pi)r to 8r, or (pi)/4
Conveniently, the ratios are both the same! :D

The difference in area is (4-pi)*r^2
The difference in perimeter is (8-2pi)*r

Square Inscribed in a circle
This one is harder to solve for, since you have to do some calculations to solve for either the radius or the side of a square.
Assuming the circle has radius r, and the square will have side r*sqrt(2).
What this means, is:
Ratio of area of circle to area of square = (pi)r^2 to 2r^2, or (pi)/2
Ratio of perimeter of circle to perimeter of square = 2(pi)r to 4r*sqrt(2), or (pi)/(2*sqrt(2))


The difference in area is (pi-2)*r^2
The difference in perimeter is (2pi - 4*sqrt(2))*r