What is pi?

a schoolyard blog asks "If someone asked you to sketch pi – not the symbol for pi, but its meaning - what would your sketch look like?"

Her math questions interest me. They make me think about how much of the math that I KNOW I really UNDERSTAND. This was a hard one - she's always asking us to DRAW a representation of things. Although I did think about this question by imagining shapes in my head, I'm not much of a sketch-er, really. Here's what I came up with:

Pi is a number but I think it's also a ratio that has to do with circles. I asked myself, okay, why is the area of a circle pi-r-squared? Thinking about that, my next question was, if I took away pi, what does r-squared represent? I imagined drawing a square around the circle so the edges of the square just touch the circle on each side. The area of that square is side-squared. The radius of the circle is half the length of one side. So, if you square just the radius, you get 1/4 of the square's area. If you multiplied by four that would give you the area of the whole square, but the circle doesn't fill the whole square. So, I decided that pi is the number that expresses what portion of the square is filled by the circle. This makes sense because pi is a bit less than four.

What do you think?