Pi is the circumference of a circle with a diameter of 1 unit. Just as a square's perimeter can be found by multiplying 4 by its width, a circle's circumference can be found by multiplying pi by its diameter. 





One can derive pi by using overlapping squares. If more and more squares geometrically overlap each other, then the perimeter of the totally overlapped area will equal ever closer to pi. 




Below shows how to calculate pi step by step using this overlapping squares method. The equation on the right is a substitution version of the equation on the bottom left of each diagram. The operation being shown is in bold. X, Y and Z are the initial variables. 

X = 1


A = 2


X = 1


B = 1.414213562...


X = 1


C = 0.414213562...


X = 1


D = 1


X = 1


Y = 0.414213562...


X = 1


Z = 8


X = 1


Pi » 3.313708498...


X = 1


E = 0.585786437...


X = 1


F = 0.292893218...


X = 1


G = 0.707106781...


X = 1


H = 0.207106781...


X = 1


I = 0.414213562...


X = 0.585786437...


X = 0.585786437...


X = 0.585786437...


A = 1.171572875...


X = 0.585786437...


B = 1.082392200...


X = 0.585786437...


C = 0.082392200...


X = 0.585786437...


D = 2.414213562...


X = 0.585786437...


Y = 0.198912367...


X = 0.585786437...


Z = 16


X = 0.585786437...


Pi » 3.182597878...


X = 0.585786437...


E = 0.801087632...


X = 0.585786437...


F = 0.400543816...


X = 0.585786437...


G = 0.599456183...


X = 0.585786437...


H = 0.240108467...


X = 0.585786437...


I = 0.480216935...


X = 0.519783064...


X = 0.519783064...


X = 0.519783064...


A = 1.039566129...


X = 0.519783064...


B = 1.019591158...


X = 0.519783064...


C = 0.019591158...


X = 0.519783064...


D = 5.027339492...


X = 0.519783064...


Y = 0.098491403...


X = 0.519783064...


Z = 32


X = 0.519783064...


Pi » 3.151724907...
