Regardless of the circle's size, this ratio will always equal pi. In decimal form, the value of pi is approximately 3. To only 18 decimal places, pi is 3. Hence, it is useful to have shorthand for this ratio of circumference to diameter. Try a brief experiment: Using a compass, draw a circle. Take one piece of string and place it on top of the circle, exactly once around. Now straighten out the string; its length is called the circumference of the circle.
Measure the circumference with a ruler. Next, measure the diameter of the circle, which is the length from any point on the circle straight through its center to another point on the opposite side. The diameter is twice the radius, the length from any point on the circle to its center.
If you divide the circumference of the circle by the diameter, you will get approximately 3. A larger circle will have a larger circumference and a larger radius, but the ratio will always be the same. If you could measure and divide perfectly, you would get 3. There are many methods to calculate Pi but I will go over the simplest to understand.
It starts with the inverse tangent function. No, you can't just plug it into your calculator and get Pithat assumes you already know Pi. Instead, we need to do a Taylor Series expansion of the inverse tangent.
The basic idea behind the Taylor Series is that any function sort of looks like a power series if you just focus on one part of that function. Using this, I can represent the inverse tangent of some value x as an infinite series:.
That's it. Now you can just plug away at this formula for as long as you likeor you could have a computer do it. Here is a program that calculates the first 10, terms in the series just press play to run it :. View Iframe URL. See, that's not so difficult for a computer. However, you can see that even after 10, terms the calculated value is still different than the accepted value.
This isn't the best series to calculate Pibut I said that earlier. This is my favorite Pi activity. Here is the idea. Generate pairs of random numbers between 0 and 1 to create random x,y coordinates. Plot these points on a 1 by 1 grid and calculate their distance to the origin. Some of these will have a origin distance less than 1 and some will be greater than 1. The points with a distance of less than one are "inside a circle"actually it's a quarter of a circle.
You really should play around with this because it's fun. Try changing the number of points or something like that. I included a "rate " statement so you can see the points being added. Oh, run it more than onceeach time you get a different result because of the random part. Get out your calculator. Use 9. Now try this:. While there is no exact value of pi, many mathematicians and math fans are interested in calculating pi to as many digits as possible. The Guinness World Record for reciting the most digits of pi belongs to Rajveer Meenaof India, who recited pi to 70, decimal places while blindfolded in Meanwhile, some computer programmers have calculated the value of pi to more than 22 trillion digits.
The website piday. Pi has been known for nearly 4, years and was discovered by ancient Babylonians. A tablet from somewhere between B. The ancient Egyptians were making similar discoveries, as evidenced by the Rhind Papyrus of B.
In this document, the Egyptians calculated the area of a circle by a formula giving pi an approximate value of 3. There is even a biblical verse where it appears pi was approximated:. And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about.
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