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Pi and its Approximations

What is pi? If this question was posed, perhaps one may be able to draw the symbolic representation of pi or rattle off the first ten digits they memorized to impress some friends in 5th grade. Maybe they think of circles or the pi day celebration they went to last year. Pi is one of the most well-known mathematical constants, representing the ratio of the circumference of a circle to its diameter (The Editors of Encyclopaedia Britannica, 2023). Pi is also irrational, meaning that it can't be represented as the ratio of any two whole numbers, and its digits don't repeat when written as a decimal. "To 39 decimal places, pi is 3.141592653589793238462643383279502884197," however an approximation such as 3.14 or 22/7 is typically used in everyday calculations (The Editors of Encyclopaedia Britannica, 2023). But how was pi discovered? And why is it so important? What is it even used for?

Check out the applet below to visualize how pi is the ratio of the circumference of a circle to its diameter!

Instructions: Move the slider to divide the line into segements whose lengths are equal to the diameter of the circle. Click "show" and use the second slider to "unroll" the circle. What does this show you?