Geometric Discovery

I was on the plane yesterday and got to thinking about circles and squares. I drew a picture:



Then I thought, "What is the relationship of the area of the outer circle to the area of the inner circle?" My gut told me it was somehow related to pi, but beyond that I really didn't know. So I set about trying to establish the relationship and discovered that the outer circle's area is exactly twice the inner. That was unexpected. Here's my proof:



This is high school geometry, nothing more. But still, I found it fascinating that the area was exactly twice. It doesn't look twice as big!