Π day

[Albert Chan]

(03-26-2022 03:59 PM)Albert Chan Wrote:  r = (b-a)/(b+b) = (1-a/b)/2 = (1-cos(x))/2 = sin(x/2)^2
x/2 = asin(sqrt(r))
...
A = sin(x)/x * pi
pi = A * (2*x/2) / (2*sin(x/2)*cos(x/2))
    = A/sqrt(1-r) * asin(sqrt(r)) / sqrt(r)
    = A/sqrt(1-r) * (1 + r/6 * (1 + r*3^2/(4*5) * (1 + r*5^2/(6*7) * (1 + r*7^2/(8*9) + ...

Above is for n-sided polygon perimeter
For n-sided polygon area, except for defintion of r, formula is exactly the same.

A = sin(2x)/(2x) * pi = sin(x)*cos(x)/x * pi
B = tan(x)/x * pi       = sin(x)/cos(x)/x * pi

r = (B-A)/B = 1-A/B = 1-cos(x)^2 = sin(x)^2

• n-sided inscribed polygon perimeter = 2n-side inscribed polygon area
• (r for n-sided polygon perimeter)     = (r for 2n-sided polygon area)

Proof for n-sided polygon perimeters implied proof of 2n-sided polygon area. QED

Note: we do not have to worry about odd-sided polygon area.
Geometrically, n required to be positive integer. But, algebraically, it does not.

From above defined A, B, this showed x = pi/n can be anything.
In other words, formula for doubling of polygon sides does not require integer sides.

A2 = sqrt(A*B) = sin(x)/x * pi

B2 = 2/(1/A2 + 1/B)
     = 2*A2 / (1 + A2/B)
     = 2*sin(x)/x * pi / (1+cos(x))
     = sin(x)/(1+cos(x)) / (x/2) * pi
     = tan(x/2) / (x/2) * pi