We calculated Pi with some of my favorite items: yarn and a tuna can. You can try it at home, too. We cut a piece of yarn that was just long enough to go around the circumference of the tuna can. Next, we straightened the yarn out and measured it with a ruler. Then, we took a piece of yarn and laid it across the top of the tuna can. That gave us its diameter. Then we did some division. If you try this at home and are still working on your long division, you can use a calculator.
Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area, and volume of a sphere. Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture, and work with them more accurately.
Historically, people had only very coarse estimations of pi such as 3, or 3. The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.
Between 3, and 4, years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are 3.
Both approximations start with 3. The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around BC, the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.
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. 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.
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