# How to Make Pi

Isaac Carstensen. 21 November, 2016.

**What is Pi?**

Pi is the circumference of a circle if the diameter of the circle is one. So if you found a line on symmetry on a circle, and the length of it was one, the length of the overall circle would be pi. This means that in terms expressed in a circle, pi = circumference/diameter. The actual value for pi is 3.14159265 to 8 decimal places, however, pi is an irrational number, meaning that it has no exact value.

**How can you Calculate Pi? **

Pi can actually be calculated in multiple different ways. The 1st way pi was discovered was by inserting a hexagon inside and outside of the circle, so each angle of the hexagon inside the circle would connect with the midpoint of each line on the outer hexagon. This method was invented by the Greek mathematician, Archimedes. This allowed him to calculate the length of each side of the inner and outer hexagon, and find the midpoint between both those values, giving an approximation to pi. This method was later improved on by adding shapes with more sides inside the circle, allowing for a more precise value to be calculated since the sides were closer to the circle. He added in more polygons until he had a 96 sided shape inside of the circle, in which he then took the approximation of pi and found that it was between 223/71 and 22/7. This was the best approximation during the Greek era, but several hundred years later a Chinese mathematician called Zu Chongzhi continued this process by adding more shapes with more sides, and he got an approximation of 355/113, which is accurate to 6 decimal places. This process was the most suitable for the time, however, as mathematics developed, it was discovered that there were multiple ways and multiple formulas that can calculate the precise value of pi.

Pi would later be calculated using infinite series, which is basically a calculation that has a sequence which continues without stop. The famous mathematician Gregory-Leibniz invented a calculation. π/4=1–1/3+1/5–1/7+1/9–1/11 etc… This equation was found by inserting the value x = 1 into the equation for inverse tan. The equation for inverse tan is tan^(-1)x=x-1/3 x³+1/5 x⁵-1/7 x⁷etc… This discovery made approximating pi very quick since there was now a constant equation that could be referred back to. This was the 1st way that pi could be expressed with a constant equation, however there were more that were later discovered, such as the Nilakantha Series (π=3+4/(2×3×4)-4/(4×5×6)+4/(6×7×8) etc…), which can be used to calculate π very accurately and very quickly, and later, much more complex ways of calculating pi were discovered. In conclusion, pi is an intriguing number that is deeply integrated into mathematics, and it shows that some of the simplest concepts in maths have the most complex solutions.

**Bibliography:**

*Institute of Mathematics: Calculating Pi*

**Further Reading:**