The value π is a constant defined as the ratio of circumference of a circle to its radius with a value of Archimedes and other mathematicians such as Ludolph Van Ceulen inscribed and circumscribed hexagons to compute the value of π. The inequality value they derived was P/(2*r) > π > P/(2*R) Further this was simplified as N x tan(180/N) > π > N x sin(180/N) The value of N starts from 6 and increases. As value of N increases, higher the accuracy of the values of the two trigonometric constants in the inequality and reaches close to exact value (3.141592653589793) as shown in table below. At 49152, the value is accuate up to 8 decimal places (3.14159265). By iterating further, we can attain the exact value. The value of π is irrational and decimal values are not repetitive (as in 22/7 = 3.132857142857...). It can be computed to required decimal places as necessary. The other method is Gregory-Leibniz Series also Madhava-Leibniz series π = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + ...... Another method is Nilakantha method as shown below. π = 3 + 4/(2 x 3 x 4) - 4/(4 x 5 x 6) + 4/(6 x 7 x 8) - 4/(8 x 9 x 10) + ...... π = 3 + 4/(24) - 4/(120) + 4/(336) - 4/(720) + 4/(1320) .... π = 3 + 0.166666 - 0.033333 + 0.011904 - 0.005555 + 0.0030303 Other way is to use trigonometric SIN(θ°) function and TAN(θ°) to compute the value of π by iteration Starting with 1°, the approximate arc length (X) with a tiny offset can be computed by slicing the radius (1 unit) to make it a right angle triangles (as shown below). Since we know the value of SIN(1), we can compute the height H. Knowing H, we can use TAN(1) function to compute R1. The angle θ shown is much larger than 1°, but makes derivation of values clearer. SIN(1) = H / Radius (R) H = SIN(1) x Radius (R) TAN(1) = H / R1 R1 = H / TAN(1) R2 = R - R1. Using Pythagoras Theorem: X = SQRT(H^{2} + R2^{2}) Using 1° to start, the circumference = 360 x X Further decreasing the angle to 0.1, 0.01, 0.001 etc. the accuracy improves (as shown in Table B above). π = Circumference / Diameter = (360 x X)/(2 x Radius) = (180 x X)/factor The Radius = 1, for 1°, the factor is 1 (360 x 1 = circumference), for 0.1°, it 10 (360 x 10 = circumference), for 0.01°, it is 100 (360 x 100 = circumference) and so on. At 1.000 degree, we get a value of 3.1415527794 with 4 decimal place accuracy At 0.100 degree, we get a value of 3.1415922548 with 6 decimal place accuracy At 0.010 degree, we get a value of 3.1415926496 with 8 decimal place accuracy At 0.001 degree, we get a value of 3.1415926535 with 10 decimal place accuracy Based on math capacity of the software used, the accuracy can be improved further by computing at 0.0001 and decreasing even further to get extremely accurate value of π. The above values in Table B are computed in Open Office, which gets best accuracy up to 10 decimal places. Reference: 1. Math Forum - Deriving Pi 2. The Institute of Mathematics 3. Math Reference
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