Verified answer

Inscribe a polygon of n sides inside a circle of radius R .- Join using two radius each side of this polygon ,. You get n triangles with angle at the center of T= 360° /n

Each triangle has a height of h= Rcos T/2 and a base B= 2Rsin(T/2) the area is

A= 2R^2 sin(T/2) cos (T/2) /2

But 2 sinT/2 cos T/2 = sin T

A= R^2 /2 sinT

The polygon has an area Ac= nA = R^2 (n/2) sin T

certainly , if n goes to infinity , the area of the polygon is close to the circle´s area .-

So , Ac = ( Lim n---> infinity ) R^2 (n/2) sinT

Ac= ( Lim n---> infinity ) R^2 (n/2) sin (360/n)

Ac~ R^2 ( Lim n---> infinity ) (n/2) sin (360/n)

The number ( Lim n---> infinity ) (n/2) sin (360/n) is PI

So Ac = pi R^2

Take a polygon of 1.000 sides , pi = 3.141571....

Now for 100.000 sides , pi = 3.141592...

As you see , pi converges quickly , happyly .-

Integration with polar coordinates...

Integrate r*dr from zero to 2pi. Done.