Step-by-step
Calculating Pi using Archimedes method of polygons.
We know that the definition of pi is the ratio of circumference of a circle divided by the diameter. Starting with Pythagorean Theorem, and proposition 3 of Euclid’s Elements, Archimedes was able to approximate pi to any precision arithmetically, without further resort to geometry!
He figured that the perimeter of any regular polygon (all sides and vertex angles equal) is an approximation to a circle. More sides will make closer approximations.
Starting with a hexagon, he bisects the central angles to make polygons 12-, 24-, 48- and 96-sides, whose perimeters approaches that of a circle, and hence the approximation to pi since the diameter remains known and constant.
Proposition 3 is also commonly referred to as the angle bisector theorem, which states that in a triangle, an angle bisector subdivides the opposite sides in the ratio of the two remaining sides.
Please refer to the attached image for the nomenclature of the geometry.
The accompanying diagram shows that the perimeter of a hexagon is 12 times the length of AB, or 12*(1.0/2) = 6. With the diameter equal to 2*1.0 = 2, the approximation to pi is 6/2=3.0.
Pi(6) = 3.0
If we divide the central angle by two, we end up with a 12-sided polygon (dodecagon), with the half central angle of 15 degrees (triangle A’BC). To calculate the new perimeter, we need to calculate the length A’B, which is given by the angle-bisector theorem as
A’B / A’A = BC / AC
All other sides are known in terms of A’B
A’B / (0.5-A’B) = sqrt(3)/2 / 1
Solve for A’B by cross-multiplication and solving for A’B, we get
A’B = sqrt(3)/(2sqrt(3)+4) = 0.2320508 (to 7 decimals)
At the same time, the radius has been reduced to
A’C = sqrt(A’B^2+BC^2) = 0.896575
That brings the approximation of pi as 12*A’B/A’C
P(12) = 3.1058285 (7 decimals)
Continuing bisecting, now using a polygon of 24 sides, we only have to replace
AB by A’B, AC by A’C, and 12 by 24 to get
Pi(24) = 3.132629 (7 decimals)
Repeating again for a polygon of 48 sides,
Pi(48) = 3.1393502
Pi(96) = 3.1410320
Pi(192) = 3.1414525
Pi(384) = 3.1415576
Etc.
The accurate value of pi to 10 digits is 3.1415926536
And we conclude that Pi(48) is the first approximation the provides 2 decimal places of accuracy.
Note: What was calculated was actually the lower bound value of pi.
We can obtain the upper bound value of pi using the length of BC as the radius, which gives the upper bound. The average of the two bounds for a 384-sided polygon gives P_mean(384) = 3.1416102, which is accurate to 2 units in the 5th decimal place.
