Deriving PiDate: 05/10/2001 at 22:02:57 From: Leslie Hannah Subject: How is Pi actually derived from the beginning to the end? I know that Pi is equivalent to 3.14 but what is the formula used to come up with the 3.14? I am aware of the way you use the formulas for a circle but could you explain to me each step of the process of deriving the formula to get 3.14? Thanks. Date: 05/11/2001 at 16:38:48 From: Doctor Rob Subject: Re: How is Pi actually derived from the beginning to the end? Thanks for writing to Ask Dr. Math, Leslie. The way that Archimedes and others up to the end of the Middle Ages used to compute Pi was to approximate it using a regular polygon of n sides and its inscribed and circumscribed circles. The inscribed circle has circumference smaller than the perimeter of the polygon, which is in turn smaller than the circumference of the circumscribed circle. That gave inequalities of the form P/(2*r) > Pi > P/(2*R) By using very large values of n, the first and last of these can be made very close together, which gives a very good estimate of Pi. These inequalities can be rewritten in terms of n, the number of sides,using trigonometric functions, as n*tan(180/n degrees) > Pi > n*sin(180/n degrees) Archimedes started with a regular hexagon, n = 6. Then 180/6 = 30 degrees is the pertinent angle, and this gives tan(30 degrees) = 1/sqrt(3), sin(30 degrees) = 1/2. This produces the inequalities 2*sqrt(3) > Pi > 3 3.464 > Pi > 3 If you double the number of sides to 12, you will cut the angle in half. You can find the tangent and sine of 15 degrees by using the formulas tan(x/2) = (sqrt[1+tan^2(x)]-1)/tan(x) sin(x/2) = sqrt[(1-sqrt[1-sin^2(x)])/2] That will give you the values tan(15 degrees) = 2 - sqrt(3) = 0.267949... sin(15 degrees) = sqrt[2-sqrt(3)]/2 = 0.258819... so 3.21539 > Pi > 3.105829 Doubling the number of sides to 24, you get tan(7.5 degrees) = 0.13165250 sin(7.5 degrees) = 0.13052619 3.15966 > Pi > 3.13263 Doubling again to 48 sides, you get tan(3.75 degrees) = 0.06540313 sin(3.75 degrees) = 0.06554346 3.14609 > Pi > 3.13935 Doubling again to 96 sides, you get tan(1.875 degrees) = 0.03273661 sin(1.875 degrees) = 0.03271908 3.14271 > Pi > 3.14103 This already shows that the first three significant figures of Pi are 3.14. This can be continued to get more and more significant figures of Pi. Ludolph Van Ceulen used this method to compute 17 decimal places of Pi in the early 1600s, which was a record at the time. To 20 decimal places, you get Pi = 3.14159265358979323846... Modern methods of computing Pi are somewhat different. This is a very complicated and interesting subject, about which I can't go into much more detail here. For more, see the following Web page from the Dr. Math Frequently Asked Questions (FAQ): About Pi http://mathforum.org/dr.math/faq/faq.pi.html - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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