I am trying to approximate $\pi$ by using inscribed and outscribed(?) polygons. I already have a formula for the circumference of the inscribed ngons:
square= √(2×4,
octagon= √(2-√(2))×8,
16gon= √(2-√(2+√(2)))×16,
32gon= √(2-√(2+√(2+√(2))))×32...
the circumference rapidly approaches tau$(2\pi)$
but im having trouble finding a formula for the outscribed 16gon. Of course I can just use a geometry program, but that doesn't help me find a generalized formula. I came up with:
sauare= 2×4
octagon= (2√(2)-2)×8
but no further. I couldn't find any formula for the outher ngons on the internet.