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,

Fig. 1

On one hand, we see the area of as

.

On the other hand, it is also

Therefore,

Or,

where by Pythagorean theorem,

Substituting (2) into (1) gives

That is,

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 :

Fig. 2

Fig. 3

*Exercise 1* Explain geometrically.

*Exercise 2* Show it is that represents