Pi! What is π? How to prove its value is approximately 3.1415?
There are more than 50 solutions found to prove phi value. They range from simple geometrical based proof to advanced Fourier series based proofs. Given below is one way, which I thought during my engineering (of-course that day the class was boring). Note that phi is irrational number. After our proof we can conclude it.
- The area of rectangle triangle OAB can be easily found which is ½ r * r.
- We left with area of the half convex formed by chord AB and perimeter of the circle. How can we calculate it? As usual, recurs.
- Draw a perpendicular OC on to AB from the origin O. Join the lines AC and BC (those in red color). By symmetry, area of the part ADC and BDC are same.
- In-order to find area of ADC and BDC, we need length of DC. We can easily calculate the lengths OD and DC. Also note that OD + OC = radius, and AD = DB = AB/2 by symmetry. We know AB from Pythagoras which is r2 + r2. From the figure we know that OD2 + BD2 = r2 (Using trigonometry also we can find OD, try it).
- After this step we can calculate OD and hence DC which is r * (SQRT(2)-1)/SQRT(2). AD and DB also known [r/SQRT(2)]. We need to repeat the above approach for pieces, ADC over the arc of circle and BDC over the arc of circle. After this step we will left with DC = r * (SQRT(2)-1)/SQRT(2), AD = r/SQRT(2), which gives us area of ∆ADC = ∆BDC = r2*(SQRT(2) - 1)/4.
- Again draw perpendiculars on to AC and CB.
- At this stage the process repeats from step 3.