Pi calculation formula series#

Finally, you would divide 1,764 by 12.57 and get 140.4. Then, you would multiply 4 by π and get 12.57. For example, if the circumference is 42 inches, first you would square 42 and get 1,764. To find the area using the circumference, or the distance around the circle, use the formula area = c^2/4π, where c is the circumference. If you havent seen the notation before it just like a sum over a for loop in python. That is quite a complicated formula, we will make more comprehensible in a moment. Therefore, the area of the circle is 314.16 inches squared. Here is Chudnovskys formula for as it is usually stated: 1 12 k 0 ( 1) k ( 6 k) ( 13591409 + 545140134 k) ( 3 k) ( k) 3 640320 3 k + 3 / 2. (Which makes sense given that the digits of Pi () go on forever. The only catch is that each formula requires you to do something an infinite number of times. Ten squared is 100, and 100 times π is 314.16. Mathematicians eventually discovered that there are in fact exact formulas for calculating Pi ().

Then, plug the radius into the formula for finding area, area = πr^2. For example, to calculate the circumference. For example, if the diameter is 20 inches, you would divide that in half and get 10 inches. Type PI(), which is equivalent to 3.14159265358979 in an Excel formula. To find the area using the diameter, or the distance from one side of the circle to the other, first divide the diameter in half to find the radius. Therefore, the area of the circle is 113.04 inches squared. Then, you would multiply 36 by π and get 113.04. For example, if the radius of the circle is 6 inches, first you would square 6 and get 36. To find the area using the radius, or the length from the center of the circle to the edge, use the formula area = πr^2, where r is the radius. Takebe calculated π to 41 decimal places with this formula.You can find the area of a circle using the radius, the diameter, or the circumference. , derived by Takebe, is the first formula to evaluate Pi in the history of Wasan.

In Wasan, Seki Takakazu, Takebe Katahiro, etc., sought calculation formulas for π 2.

In Enri shinko by Wada Yasushi was equivalent to Therefore, Kikuchi proved in the next paper that the calculation (If you compute to n terms and multiply by 4, you get within roughly 1/n of the value of so to get 3 decimal places of accuracy, you'll need to sum something like the first thousand terms of the series.) There is the Machin formula.

Pi calculation formula series#

A hypergeometric series is defined as follows: Unfortunately, this converges rather slowly. Gauss (1777-1855) named a hypergeometric series. Kikuchi noticed that such a series was what K. Each term is decided by multiplying its previous term by a regular fraction as follows: In fact, however, there is a relationship between the terms. We do not know anything about the number's regularity from this result alone. Holds true Hasegawa uses this to obtain the result of (the sum of the powers of the natural numbers), We can obtain a value of 3.14159 for π accurate to five decimal places with the first 4 terms of the Taylor expansion of tan -1.In a recent computer calculation, the following equations were used: Moreover, Newton (1642-1727) and Euler (1707-1783) discovered a series that converged faster, which enabled them to calculate values of Pi to more decimal places. In Europe, Viete (1540-1603) discovered the first formula that expresses π:Īfter that, the Wallis (1616-1703) Formula: Wasan scholars such as Muramatsu Shigekiyo, Seki Takakazu, Kamata Toshikiyo, Takebe Katahiro, and Matsunaga Yoshisuke calculated more accurate values of Pi, and accomplished results that could be compared to European mathematics. In the Edo Period of Japan, Jinkoki (1627) by Yoshida Mitsuyoshi used 3.16 for Pi, but as people recognized that this value was not accurate, a field called Enri ( en means a circle and ri means a theory), in which more accurate values for Pi were calculated, began to evolve. In ancient India, we can find an example of the use of In the ancient Egypt, they obtained an approximation ofīy placing a regular octagon on a circle, and in ancient Babylonia they usedĪrchimedes came to the conclusion in his work Kyklu metresis (measure of a circle) that Pi satisfies As a regular hexagon that is inscribed in a circle with a radius of 1 has a perimeter of 6, it is revealed that Pi has a value greater than 3. As for the value of π, ancient civilizations used their own.