# Renaldini construction

Approximated construction for π/n and regular polygon with n sides.

ABC is an equilateral triangle, AP = AB/r.
Then t ≈ π / r

cos(π / r) ≈ cos(t) = (r - 2)×( 3r + √(r² + 8r - 8) ) / ( 4(r² - r + 1) ) |

Let r = n/2 hence t = 2π/n, we build an approximate value of the side AQ of regular polygon with n sides.

Inaccuracy is for different values of n :

n |
approx 2π/n |
2π/n |
err 2π/n |
approx s_{n} |
s_{n} |
err s_{n} % |

This is not a very good accuracy, but there is a better construction than the original method.

Because the construction gives an exact value for π/3,
instead of constructing 2π/n
we can construct kπ/n ≈ π/3

Which results into the construction of a star polygon,
from which we deduce the corresponding convex polygon.

For instance n = 19 : k/19 ≈ 1/3 gives k = 6 that is AP/AB = 1/r = 6/19.

t = 18° 57' 23", error 0° 0' 32", QQ' = 0.329346, error = 0.05%,
to compare with previous value 3.3%.

Another better way is to construct an angle at nearest from on other exact value π/2 :

#### From the center

ABC is again an equilateral triangle and OP = AB/n.

Then t ≈ π / n
and QQ' an approximate value of the side of regular n-gon.

sin(π / n) ≈ sin(t) = (6n + 2√(3n² - 8) ) / (3n² + 4) |

and related inaccuracies :

n |
approx 2π/n |
2π/n |
err 2π/n |
approx s_{n} |
s_{n} |
err s_{n} % |

Which is much better, without the need of a star polygon.