This post illustrates an alternative of compute the approximate value of .
We begin with a circle whose radius is , and let denotes the side’s length of regular polygon inscribed in the circle with and sides respectively,
On one hand, we see the area of as
On the other hand, it is also
where by Pythagorean theorem,
Substituting (2) into (1) gives
Let , we have
Solving (3) for yields
Since must be greater than (see Exercise 1), it must be true (see Exercise 2) that
Notice when , we obtain (5) in “Truth vs. Intellect“.
With increasing ,
We can now compute the approximate value of from any circle with radius :
Exercise 1 Explain geometrically.
Exercise 2 Show it is that represents