How to find pi?

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I'm a novice here, but have always wondered this... I can divide 1 by 3 to get the recurring decimal 0.3333... But how would I find pi? I could take the circumference of a giant circle and compare it to its diameter, or use the pi button on a calculator, but when someone figures out pi to the 10,000th digit or so, how do they know they've got it right? Is there a simple algorithm (such as using long division to compute 1/3) that laymen can use to explore the digits of pi?