How can I plot a polygon with n sides inscribed in a circle and show the area?

Question

I am doing some work on Archimedes and want to show what the area of a regular n-sided polygon is within a circle. My professor from two years ago was able to show it with an adjustable slider that increased the number of sides of a polygon. I was wondering if it's possible to tack on an equation to display the area of the polygon.

I'm not very good at plotting with Mathematica and need help with all of the code.

Answer 1

Off[Solve::ztest];
var = {R, r, a, p, s};
assume = Join[
   Thread[var > 0],
   {R > r, Element[n, Integers], n > 2}];
eqns = {
   R == s*Csc[Pi/n]/2,
   r == s*Cot[Pi/n]/2,
   a == n*s^2*Cot[Pi/n]/4,
   p == n*s};

sol = Reverse[Assuming[assume,
     Simplify[Solve[
          Join[eqns, assume], #, Reals][[1]] & /@
       Select[
        Subsets[var, {Length[var] - 1}],
        MemberQ[#, a] &]]]] // FullSimplify;

Manipulate[
 If[IntegerQ[m], v = m, v = 5];
 pts = Table[{Cos[2 Pi*k/v], Sin[2 Pi*k/v]},
   {k, 0, v}];
 Column[{
   Graphics[{
     AbsoluteDashing[8],
     Lighter[Gray, 0.4],
     Circle[],
     If[Not[m === Infinity],
      {Circle[{0, 0}, Cos[Pi/v]],
       Blue,
       Line[{{0, 0}, #}] & /@ Take[pts, 2],
       Text["R", {1/2, 1}*pts[[1]], {0, 2}],
       Magenta, Text["\[Alpha]", Plus @@ Take[pts, 2]/7],
       Text["\[Beta]", 6/7*pts[[3]]],
       Darker[Green, .25],
       AbsoluteDashing[4],
       Line[{{0, 0}, Cos[Pi/v]
          {Cos[5 Pi/v], Sin[5 Pi/v]}}],
       Text["r", Cos[Pi/v]/2
         {Cos[1.1*5 Pi/v], Sin[1.1*5 Pi/v]}]}],
     Dashing[{}],
     Black,
     Thick,
     If[m === Infinity,
      {Circle[{0, 0}]},
      Line[pts]]}],
   Grid[
     ReplacePart[arr = Transpose[
        Join[
         {Style[#, Darker[Blue]] & /@
           {"circumradius", 
            "inradius", "area",
            "perimeter", "edge length",
            "central\nangle, \[Alpha]",
            "interior\nangle, \[Beta]",
            "interior\nangle sum"}},
         tab = (TraditionalForm /@ Join[var,
              {If[m === Infinity, 0,
                Row[{2 Pi/n, " rad", ", i.e., ",
                  FunctionExpand[2 Pi/n/Degree] Degree}]]},
              {If[m === Infinity, "\[Pi] rad, i.e., 180\[Degree]",
                Row[{(n - 2) Pi/n, " rad", ", i.e., ",
                  FunctionExpand[(n - 2) Pi/n/Degree] Degree}]]},
              {If[m === Infinity, "Indeterminate", 
                Row[{(n - 2) Pi, " rad", ", i.e., ",
                  FunctionExpand[(n - 2) Pi/Degree] Degree}]]}]) /.

                    If[m === Infinity,
            Map[#[[1]] -> Limit[#[[-1]] /. s -> 2 R Pi/n,
                n -> Infinity] &, sol, {2}],
            sol]]],
      Flatten[Table[{i, j} -> SpanFromLeft,
        {i, If[m === Infinity, 5, 6], 8},
        {j, 3, 5}]]],
     Frame -> All,
     Alignment -> {Center, Center}] /. n -> m},
  Alignment -> Center],
 {{m, "n", "number of edges"},
  Join[{"n"}, Range[3, 20], {Infinity}],
  ControlType -> SetterBar}]

Answer 2

Manipulate[{"Area =" <> ToString[N@n Cos[Pi/(2 n)] Sin[Pi/(2 n)]], 
  Graphics[{Circle[], {Yellow, Polygon[CirclePoints[n]]}}]}, {n, 3, 
  50, 1}]

Answer 3

I understand the down voting for this question. However, for fun:

p[n_] := {Cos[2 Pi #/n], Sin[2 Pi #/n]} & /@ Range[0, n]
piapp[n_] := With[{pg = Polygon[p[n]]},
  Graphics[{Circle[], Red, pg, 
    Text[Style[N@n Sin[2 Pi/n]/2, White, 16], {0, 0}]}, 
   ImageSize -> 300]]
anim = Table[
   Row[{
     piapp[num],
     ListPlot[Table[{j, N@j Sin[2 Pi/j]/2}, {j, Range[3, 100]}], 
      GridLines -> {None, {Pi}}, Joined -> True, 
      Epilog -> {Red, PointSize[0.02], 
        Point[{num, N@num Sin[2 Pi/num]/2}], 
        Text[Style[num, 
          12], {num, N@num Sin[2 Pi/num]/2} - {-5, 0.25}]}, 
      PlotRange -> {0, 3.2}, ImageSize -> 300]}], {num, 
    Range[3, 100]}];

The Table command can be changed to Manipulate. It was used to make animated gif.