Squaring the Circle

While staggering around my local lanes recovering from a bad back I had injured jumping out of our apple tree to my wife's annoyance, I got to thinking about circles and squares. A few months ago I had spent a considerable amount of time trying to establish the area of a circle as recorded here.

This is a lot simpler, "Is there a relationship between the area of a square inscribed and circumscribed on a circle?"  A much easier problem quite capable of being resolved whilst out walking.

Calculation

Inscribed square MNPQ

Area = QM x MN = QM2

^QOM = 1 rt angle  

  OM = OQ = r (radius of circle)

So

 QM2 = 2r2 

Circumscribed square ABCD

Area = AB2 = (2r)2 = 4r2

Relationships

Area Outer : Inner = 2:1

Area Outer : Circle : Inner 

=  4 : π  : 2