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 sn sn err sn %

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 sn sn err sn %

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

 

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