Pi is DEFINED as the ratio of the radius to the circumference. The only question is: what is the numerical value of that ratio. Since it is NOT a rational ratio, it can only be estimated.

There are some VERY powerful formulas for estimating Pi, i.e. each iteration corrects the last one of two digits and adds on five or six more. I have not seen visual analogues that are so powerful, but there do exist less powerful version. Try this as a starter:

Put a square with sides 2r around the circle and another one with sides r inside it. The perimeter of the large square is 8r and that of the small one, 4r, so 2Pi is between 8 and 4, or Pi is between 4 and 2. How's that for a "first-order" approximation!

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