Area of any n equal-sided polygon

I remember deriving this formula in high school.
Let:
n = number of sides
s = length of a side

Area of Polygon = (n(s^2))/(4tan(180/n))

To check quickly, for a square,
n = 4
tan(45) = 1

Area of a square = (4(s^2))/(4(1)) = s^2

It is interesting to note that the Area of a Polygon approximates the Area of a Circle when n approaches a very large value, say infinity. But as n approaches a very large value, the value of s approaches zero since the length of a side almost becomes a point. We have to modify our Area of a Polygon formula so that we represent the length of a side with the polygon's apothem. (And as n approaches infinity, this is also the radius of a circle).

a = apothem length = radius of a circle
s = a(2(tan(180/n))

Area of Polygon = n(a^2)(tan(180/n)

In this formula, as n approaches infinity, then the formula approximates the area of a circle.

Area of a Circle = π(a^2)

So, π can be approximated.

π = n(tan(180/n))

You can try out any number which is relatively larger than 180, say 100,000 and compute for the value of π. When n =100,000, π = 3.14159.

It is also interesting for me that (π/n) = tan (180/n). From trigonometry, we know that tan(Θ) = Opposite/Adjacent. tan(180/n) = π/n. So we have a right triangle with π for the opposite side and n for the adjacent side. The term (180/n) is actually (360/2n). We are dividing 360 degrees by twice the value of n. As n approaches infinity, the angle approaches zero and we get the value of π.