Sia (Subscript[P, n]) un poligono regolare di n lati.

1) Qual è il minimo valore di n tale che il rapporto tra il perimetro di (Subscript[P, n]) ed il diametro della circonferenza circoscritta sia maggiore di 3.14?

TEST: Indicare se 
	a) n < 20
	b) 20 ≤ n < 30
	c) 30 ≤ n < 40
	d) 40 ≤ n < 50
	e) 50 ≤ n < 60
	f) 60 ≤ n < 100
	g) n ⩾ 100

2) Qual è il minimo valore di n tale che (Subscript[P, n])  non sia distinguibile da un cerchio?

?Graphics

Graphics[Point[{0,0}]]

Graphics[Point[{0,0}],Axes->True]

Graphics[{Red,PointSize[.02],Point[{0,0}]},Axes->True]

Graphics[{PointSize[.02],Point[{0,0}],Red,Point[{1,0}]},Axes->True]

?PlotRange

Options[Graphics]

Options[Plot]

GoldenRatio

N[GoldenRatio]

N[GoldenRatio,20]

N[GoldenRatio,200]

N[GoldenRatio,2000]

Graphics[{PointSize[.02],Point[{0,0}],Red,Point[{1,0}]},Axes->True,PlotRange->{{-1,1},{-1,1}}]

Table[k,{k,1,10}]

Table[k,{k,1,10,1/2}]

Table[{Cos[2k*Pi/3],Sin[2k*Pi/3]},{k,1,3}]

Table[{Cos[2k*Pi/5],Sin[2k*Pi/5]},{k,1,5}]

Table[{Cos[2k*Pi/7],Sin[2k*Pi/7]},{k,1,7}]

2^(2^4)+1

Graphics[
{PointSize[.02],Red,
Point[
Table[{Cos[2k*Pi/7],Sin[2k*Pi/7]},{k,1,7}]
]
},
Axes->True,PlotRange->{{-1,1},{-1,1}},ImageSize->300
]

Manipulate[
Graphics[
{PointSize[.02],Red,
Point[
Table[{Cos[2k*Pi/n],Sin[2k*Pi/n]},{k,1,n}]
]
},
Axes->True,PlotRange->{{-1,1},{-1,1}},ImageSize->200
],
{n,3,50,1}]

Manipulate[
Graphics[
{Thickness[.005],Red,
Line[
Table[{Cos[2k*Pi/n],Sin[2k*Pi/n]},{k,0,n}]
]
},
Axes->True,PlotRange->{{-1,1},{-1,1}},ImageSize->200
],
{n,3,60,1}]

Manipulate[Graphics[{{Red,Circle[{0,0},1]},Line[Table[{Cos[k*2Pi/n],Sin[k*2Pi/n]},{k,0,n,1}]]},ImageSize->400],{n,3,100,1}]

Manipulate[Graphics[{Line[Table[{Cos[k*2Pi/n],Sin[k*2Pi/n]},{k,0,n,1}]]},ImageSize->400],{n,3,100,1}]

(* (Subscript[ℓ, n])=2r*sin((π/n)) , (Subscript[π, n])=n*((Subscript[ℓ, n])/(2r))=n*sin((π/n)) *)

Pi

Sin[Pi/3]

N[Pi,3]

N[Pi]

N[Pi,20]

N[Pi,2000]

Table[n*Sin[Pi/n],{n,3,10}]

ToRadicals[Sin[Pi/8]]

Sin[Pi/8]

ToRadicals[Sin[Pi/7]]

Table[N[n*Sin[Pi/n]],{n,3,10}]

Table[{n,N[n*Sin[Pi/n]]},{n,3,10}]

Table[{n,N[n*Sin[Pi/n]]},{n,3,60}]

Table[{n,N[n*Sin[Pi/n],10]},{n,3,60}]

TableForm[Table[{n,N[n*Sin[Pi/n],10]},{n,3,60}]]

costr[n_]:=Block[{com=Complement[FactorInteger[n],{{3,1},{5,1},{17,1},{257,1},{65537,1}}],res},
res=If[Length[com]==0||Length[com]==1&&com[[1,1]]==2,True,False,False];
Return[res];
];

TableForm[Table[{Style[n,If[costr[n],FontColor->Red,FontColor->Black]],N[n*Sin[Pi/n],7],If[costr[n],Simplify[ToRadicals[n*Sin[Pi/n]]],""]},{n,3,50}],TableHeadings->{None,{"n","(Subscript[π, n])","(Subscript[π, n]) per radicali"}}]

TableForm[Table[{n,N[n*Sin[Pi/n],7]},{n,3,60,1}],TableHeadings->{None,{"n","(Subscript[π, n])"}}]

appk[k_]:=Block[{n=3},While[n*Sin[Pi/n]<ToExpression[StringJoin[Drop[Characters[ToString[N[Pi,k+1]]],-1]]],n++];Return[n];];

TableForm[Table[{p,appk[p]},{p,1,10,1}]]

Timing[TableForm[Table[{p,appk[p]},{p,1,10,1}]]]