Regular Pentagon

Author

Eric W. Weisstein
July 3, 2018

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polygons/RegularPentagon.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RegularPentagon.html.

©2018 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=Sqrt[10] (2 a+5 x)≥15 Sqrt[2] x+2 Sqrt[25-5 Sqrt[5]] y&&2 Sqrt[10] a≥5 Sqrt[2] (-3+Sqrt[5]) x+2 Sqrt[25-5 Sqrt[5]] y&&2 Sqrt[10] a+5 Sqrt[2] (-1+Sqrt[5]) x+2 Sqrt[125-55 Sqrt[5]] y≥0&&Sqrt[3+Sqrt[5]] a+2 Sqrt[5-Sqrt[5]] y≥0&&2 Sqrt[10] a+2 Sqrt[125-55 Sqrt[5]] y≥5 Sqrt[2] (-1+Sqrt[5]) x;

implreg=ImplicitRegion[ineq,{x,y}];

assum=a>0;

verts=a{{1/4 (1+Sqrt[5]),Root[1-20 #1^2+80 #1^4&,3]},{0,Sqrt[1/10 (5+Sqrt[5])]},{1/4 (-1-Sqrt[5]),Root[1-20 #1^2+80 #1^4&,3]},{-(1/2),-(1/2) Sqrt[1+2/Sqrt[5]]},{1/2,-(1/2) Sqrt[1+2/Sqrt[5]]}};

reg=Polygon[verts];

Figure

Show[LaminaData["FilledRegularPentagon","Diagram"],ImageSize->300]

Plots

Diagram

LaminaData["FilledRegularPentagon","Diagram"]

DiscretizeRegion

Block[{a=1},DiscretizeRegion[#]]&/@{reg,implreg}

Polygons

Graphics[RegularPolygon[5,1,π/10],ImageSize->Small]

Vertex Coordinates

Vertex at (1, 0)

Cos[Pi/5]

Sin[2Pi/5]

Abs[Table[{Cos[#],Sin[#]}&[2k Pi/5],{k,0,4}]]//ColumnForm

Vertex at (0, 1)

Abs[Table[{Sin[#],Cos[#]}&[2k Pi/5],{k,0,4}]]//ColumnForm

Properties

FullSimplify[RegularPolygonInformation[5]//TableForm//TraditionalForm,ExcludedForms->{Degree}]

Sagitta

FullSimplify[1/2 Tan[Pi/10]]

N[%]

(1/10) (Sqrt[25-10Sqrt[5]])//N

Diagonal

Solve[((Sqrt[(d-x)^2-(d/2)^2])/(Sqrt[x^2-(1/2)^2]))==((d/2)/(1/2))==((d-x)/x),d]

Properties (point-up orientation)

Vertices

FullSimplify[Through[{MinValue,MaxValue}[{x,ineq[x,y]},{x,y}]],a>0]

FullSimplify[Through[{MinValue,MaxValue}[{y,ineq[x,y]},{x,y}]],a>0]

Inequality

RegularPolygon[5,1/2Csc[π/5],π/10]//FullSimplify

ineq=FullSimplify[Inequalities[RegularPolygon[5,a/2Csc[π/5],π/10],{x,y}],a>0]

ineq[x_,y_]:=(Sqrt[10]) (2 a+5 x)≥15 (Sqrt[2]) x+2 (Sqrt[25-5 (Sqrt[5])]) y&&2 (Sqrt[10]) a≥5 (Sqrt[2]) (-3+(Sqrt[5])) x+2 (Sqrt[25-5 (Sqrt[5])]) y&&2 (Sqrt[10]) a+5 (Sqrt[2]) (-1+(Sqrt[5])) x+2 (Sqrt[125-55 (Sqrt[5])]) y≥0&&(Sqrt[3+(Sqrt[5])]) a+2 (Sqrt[5-(Sqrt[5])]) y≥0&&2 (Sqrt[10]) a+2 (Sqrt[125-55 (Sqrt[5])]) y≥5 (Sqrt[2]) (-1+(Sqrt[5])) x

RegionPlot[ineq[x,y]/.a->1,{x,-1,1},{y,-1,1}]

Area

Area

Assuming[assum,FullSimplify[Area[reg]]]

Assuming[assum,FullSimplify[Area[reg/.a->1]]]

Assuming[assum,FullSimplify[Area[implreg]]]

Integrate

Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing

RegionMeasure

Assuming[assum,FullSimplify[RegionMeasure[reg]]]

Assuming[assum,FullSimplify[RegionMeasure[reg/.a->1]]]

Assuming[assum,FullSimplify[RegionMeasure[implreg]]]

AreaInertiaTensor

Assuming[a>0,AreaInertiaTensor[ineq[x,y],{x,y}]]//FullSimplify//Timing

Centroid

Integrate

Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing

RegionCentroid

Assuming[assum,FullSimplify[RegionCentroid[reg]]]

Assuming[assum,FullSimplify[RegionCentroid[reg/.a->1]]]

Assuming[assum,FullSimplify[RegionCentroid[implreg]]]

Convex

TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing

TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing

Diagonals

Length[v={{1/4 (1+Sqrt[5]),Root[1-20 #1^2+80 #1^4&,3]},{0,Sqrt[1/10 (5+Sqrt[5])]},{1/4 (-1-Sqrt[5]),Root[1-20 #1^2+80 #1^4&,3]},{-(1/2),-(1/2) Sqrt[1+2/Sqrt[5]]},{1/2,-(1/2) Sqrt[1+2/Sqrt[5]]}}]

ds=With[{n=Length[v]},Select[Subsets[Range[n],{2}],Mod[Subtract@@#,n]≠n-1≠1&]]

Length[%]

FullSimplify[Binomial[n,2]-n]

n(n-3)/2/.n->Length[v]

Graphics[{
{Yellow,Opacity[.2],EdgeForm[Black],Polygon[v]},
Line[ds]/.Thread[Range[Length[v]]->v]
}]

SortBy[FullSimplify@RootReduce[Norm[Subtract@@v[[#]]]&/@ds],N]/.(1/2) (1+(Sqrt[5]))->GoldenRatio

EdgeLengths

s=FullSimplify[Norm/@(Subtract@@@Partition[verts,2,1,1]),assum]

GeneralizedDiameter

FullSimplify[Sqrt[a^2((((1/2)Csc[(π/n)]+(1/2)Cot[(π/n)]))^2)+(1/4)(a^2)],a>0&&n>0]

GD[n_]:=If[OddQ[n],(1/2) Csc[(π/(2 n))],Csc[(π/n)]]a

Table[GD[n],{n,3,8}]//FunctionExpand//FullSimplify

ineqs=FullSimplify[And@@ineq@@@{{x1,y1},{x2,y2}},a>0]/.a->1

Maximize[{(x1-x2)^2+(y1-y2)^2,ineqs},{x1,x2,y1,y2},Reals]//Timing

Perimeter

Total[s]//FullSimplify

RadiiOfGyration

Assuming[a>0,FullSimplify[RadiiOfGyration[ineq[x,y],{x,y}]]]//Timing

Properties (point-right orientation)

Inequality

FullSimplify[Inequalities[RegularPolygon[5,1/2Csc[π/5]],{x,y}],a>0]

ineq[x_,y_]:=((Sqrt[10]) (2+5 y)≥2 (Sqrt[25-5 (Sqrt[5])]) x+15 (Sqrt[2]) y&&2 (Sqrt[10])+2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) y≥5 (Sqrt[10]) y&&(Sqrt[2])+(Sqrt[10])+4 (Sqrt[5-(Sqrt[5])]) x≥0&&2 (Sqrt[10])+2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) (-1+(Sqrt[5])) y≥0&&2 (Sqrt[25-5 (Sqrt[5])]) x+5 (Sqrt[2]) (-3+(Sqrt[5])) y≤2 (Sqrt[10]))

RegionPlot[ineq[x,y],{x,-1,1},{y,-1,1}]

Area

Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing

Centroid

Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing

InertiaTensor

AreaInertiaTensor[ineq[x,y],{x,y}]//FullSimplify//Timing

Properties (arbitrary orientation)

Inequality

FullSimplify[Inequalities[RegularPolygon[5,1/2Csc[π/5],q],{x,y}],a>0&&0≤q<2π]

ineq[x_,y_]:=2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x+5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(-5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]≤5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]+2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&(((-1))^(1/10)) (-1+(((-1))^(2/5))) ((1+(((-1))^(2/5))) (Sqrt[5-(Sqrt[5])])-(((-1))^(1/5)) (Sqrt[2]) (-5+(Sqrt[5])) x Cos[q]-(((-1))^(1/5)) (Sqrt[2]) (-5+(Sqrt[5])) y Sin[q])≤0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x-5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]+5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]≤2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x-5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0

Block[{q=0},RegionPlot[Chop[ineq[x,y]],{x,-1,1},{y,-1,1}]]

Block[{q=.2},RegionPlot[Chop[ineq[x,y]],{x,-1,1},{y,-1,1}]]

simplifies

FullSimplify@ExpToTrig[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q]) <= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0]

ineq2[x_,y_]:=2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x+5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(-5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]≤5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]+2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&1+2 (Sqrt[5-2 (Sqrt[5])]) (x Cos[q]+y Sin[q])≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x-5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]+5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]≤2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x-5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0

Block[{q=.0},RegionPlot[Chop[ineq2[x,y]],{x,-1,1},{y,-1,1}]]

Area

Assuming[0≤q<2π,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing

Exact

N[(1/4) (Sqrt[5 (5+2 (Sqrt[5]))])]

With[{q=0},Integrate[Boole[2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x+5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(-5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]≤5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]+2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&1+2 (Sqrt[5-2 (Sqrt[5])]) (x Cos[q]+y Sin[q])≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x-5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]+5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]≤2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x-5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0], {x, -1, 1}, {y, -1, 1}]]//FullSimplify//Timing

N[%[[2]]]

QuasiMonteCarlo

With[{q=0},NIntegrate[Boole[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q]) <= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0], {x, -1, 1}, {y, -1, 1}, Method->"QuasiMonteCarlo"]]//Timing

...wrong...

With[{q=0},NIntegrate[Boole[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && Re[
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q])] <= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0], {x, -1, 1}, {y, -1, 1},Method->"QuasiMonteCarlo"]]//Timing

Default

With[{q=0},NIntegrate[Boole[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q]) <= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0], {x, -1, 1}, {y, -1, 1}]]//Timing

...wrong...

With[{q=0},NIntegrate[Boole[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && Re[
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q]) ]<= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0], {x, -1, 1}, {y, -1, 1}]]//Timing

MaxPrecision

With[{q=0},NIntegrate[Boole[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q]) <= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0], {x, -1, 1}, {y, -1, 1}, WorkingPrecision->20]]//Timing

...wrong...

With[{q=0},NIntegrate[Boole[2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x + 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (-5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0 && 
    (-2*Sqrt[125 - 55*Sqrt[5]]*x + 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] <= 5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] + 
      2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && Re[
    (-1)^(1/10)*(-1 + (-1)^(2/5))*((1 + (-1)^(2/5))*Sqrt[5 - Sqrt[5]] - (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*x*Cos[q] - 
       (-1)^(1/5)*Sqrt[2]*(-5 + Sqrt[5])*y*Sin[q]) ]<= 0 && (-2*Sqrt[125 - 55*Sqrt[5]]*x - 5*Sqrt[2]*(-1 + Sqrt[5])*y)*Cos[q] + 
      5*Sqrt[2]*(-1 + Sqrt[5])*x*Sin[q] <= 2*(Sqrt[10] + Sqrt[125 - 55*Sqrt[5]]*y*Sin[q]) && 
    2*Sqrt[10] + (-2*Sqrt[25 - 5*Sqrt[5]]*x - 5*Sqrt[2]*(-3 + Sqrt[5])*y)*Cos[q] + 
      (5*Sqrt[2]*(-3 + Sqrt[5])*x - 2*Sqrt[25 - 5*Sqrt[5]]*y)*Sin[q] >= 0], {x, -1, 1}, {y, -1, 1}, WorkingPrecision->20]]//Timing

Simplified symbolic

With[{q=0},NIntegrate[Boole[2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x+5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(-5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]≤5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]+2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&1+2 (Sqrt[5-2 (Sqrt[5])]) (x Cos[q]+y Sin[q])≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x-5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]+5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]≤2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x-5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0], {x, -1, 1}, {y, -1, 1}]]//Timing

Do[Print[{q,NIntegrate[Boole[2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x+5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(-5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x+5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]≤5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]+2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&1+2 (Sqrt[5-2 (Sqrt[5])]) (x Cos[q]+y Sin[q])≥0&&(-2 (Sqrt[125-55 (Sqrt[5])]) x-5 (Sqrt[2]) (-1+(Sqrt[5])) y) Cos[q]+5 (Sqrt[2]) (-1+(Sqrt[5])) x Sin[q]≤2 ((Sqrt[10])+(Sqrt[125-55 (Sqrt[5])]) y Sin[q])&&2 (Sqrt[10])+(-2 (Sqrt[25-5 (Sqrt[5])]) x-5 (Sqrt[2]) (-3+(Sqrt[5])) y) Cos[q]+(5 (Sqrt[2]) (-3+(Sqrt[5])) x-2 (Sqrt[25-5 (Sqrt[5])]) y) Sin[q]≥0], {x, -1, 1}, {y, -1, 1}]//Timing}],{q,0.,2π,.1}]

Centroid

Assuming[0≤q<2π,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing

InertiaTensor

Assuming[0≤q<2π,FullSimplify[AreaInertiaTensor[ineq[x,y],{x,y}]]]//Timing

Lengths

in=ineq[x1,y1]&&ineq[x2,y2]/.a->1

NIntegrate

NIntegrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[in],{x1,-1,1},{y1,-1,1},{x2,-1,1},{y2,-1,1}]

NIntegrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[in],{x1,-1,1},{y1,-1,1},{x2,-1,1},{y2,-1,1},Method->"QuasiMonteCarlo",MaxPoints->10^8,PrecisionGoal->2]//Timing

minx=-(1/4) (1+(Sqrt[5])) ;
maxx=(1/4) (1+(Sqrt[5])) ;
miny=-(1/2) (Sqrt[1+(2/(Sqrt[5]))]) ;
maxy=(Sqrt[(1/10) (5+(Sqrt[5]))]) ;

NIntegrate[Boole[in],{x1,minx,maxx},{y1,miny,maxy},{x2,minx,maxx},{y2,miny,maxy},Method->"QuasiMonteCarlo"]

((((1/4) (Sqrt[5 (5+2 (Sqrt[5]))])))^2)//N

NIntegrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[in],{x1,minx,maxx},{y1,miny,maxy},{x2,minx,maxx},{y2,miny,maxy},Method->"QuasiMonteCarlo",MaxPoints->10^8,PrecisionGoal->2]//Timing

NIntegrate[(Sqrt[(x1-x2)^2+(y1-y2)^2])Boole[in],{x1,minx,maxx},{y1,miny,maxy},{x2,minx,maxx},{y2,miny,maxy},Method->"QuasiMonteCarlo",MaxPoints->10^8,PrecisionGoal->3]//Timing

...looks close to 2...

NIntegrate[(Sqrt[(x1-x2)^2+(((y1-y2))^2)])Boole[in],{x1,minx,maxx},{y1,miny,maxy},{x2,minx,maxx},{y2,miny,maxy},Method->"QuasiMonteCarlo",MaxPoints->10^8,PrecisionGoal->4]//Timing

...ran overnight...

4-D Integral

P[0]={0,0};
{P[1],P[2],P[3],P[4],P[5]}=Table[{Sin[φ],Cos[φ]}/Sqrt[(1/2)*(5-Sqrt[5])],{φ,0,2 Pi 4/5,2 Pi/5}]//Simplify;

distanceTable=Table[Print[j];
{x1,y1}=P[0]+t1 (1-s1) P[1]+(1-t1) (1-s1) P[2];
{x2,y2}=P[0]+t2 (1-s2) P[j]+(1-t2) (1-s2) P[Mod[j+1,5,1]];
det=Det[Outer[D,{x1,y1,x2,y2},{s1,t1,s2,t2}]]//Simplify;
dist=Sqrt[(x1-x2)^2+(y1-y2)^2]//FullSimplify;
NIntegrate[Evaluate[dist det],{s1,0,1},{t1,0,1},{s2,0,1},{t2,0,1},WorkingPrecision->10],{j,5}]

5*Total[distanceTable]

distanceTable=Table[Print[j];
{x1,y1}=P[0]+t1 (1-s1) P[1]+(1-t1) (1-s1) P[2];
{x2,y2}=P[0]+t2 (1-s2) P[j]+(1-t2) (1-s2) P[Mod[j+1,5,1]];
det=Det[Outer[D,{x1,y1,x2,y2},{s1,t1,s2,t2}]]//Simplify;
dist=Sqrt[(x1-x2)^2+(y1-y2)^2]//FullSimplify;
Integrate[Evaluate[dist det],{s1,0,1},{t1,0,1},{s2,0,1},{t2,0,1}],{j,1}]

Integrate[Evaluate[dist det],{s1,0,1},Assumptions->0<t1<1&&0<s2<1&&0<t2<1]

Integrate[Evaluate[dist det],s1]

i1=(1/(40 (Sqrt[2])))(5+2 (Sqrt[5])) (-1+s2) (1/(3 (Sqrt[2]) (((-1+t1))^2))(4 (((s2+s1 (-1+t1)-t1+t2-s2 t2))^2)-((-1+s2) (s1+t1-s1 t1+s2 (-1+t2)-t2) (12-(11+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t1^2) (-7+6 t2)+t1 (-31+7 (Sqrt[5])-5 (-5+(Sqrt[5])) t2)))/(-2-(-5+(Sqrt[5])) t1+(-5+(Sqrt[5])) (t1^2))+((((-1+s2))^2) (t1-t2) (-6+6 (Sqrt[5])-t2-9 (Sqrt[5]) t2+(t1^2) (-130+38 (Sqrt[5])+(125-55 (Sqrt[5])) t2)+(t1^3) (85-23 (Sqrt[5])+30 (-3+(Sqrt[5])) t2)+t1 (67-21 (Sqrt[5])+(-50+34 (Sqrt[5])) t2)))/(((2+(-5+(Sqrt[5])) t1-(-5+(Sqrt[5])) (t1^2)))^2)) √(1/(-5+(Sqrt[5]))((s1^2) (-2-(-5+(Sqrt[5])) t1+(-5+(Sqrt[5])) (t1^2))+(-5+(Sqrt[5])) (((t1-t2))^2)+(-5+(Sqrt[5])) s2 (t1-t2) (-1+2 t2)+(s2^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2))+s1 (-(-5+(Sqrt[5])) (-1+2 t1) (t1-t2)+s2 (4+(-5+(Sqrt[5])) t2-(-5+(Sqrt[5])) t1 (-1+2 t2)))))+(I (((-1+s2))^3) (((t1-t2))^2) (5+(Sqrt[5])-5 t2+5 t1 (-1+2 t2)) Log[(4 (-1+t1) (-10 I s1+2 I (Sqrt[5]) s1+10 I s2-2 I (Sqrt[5]) s2-15 I t1+5 I (Sqrt[5]) t1+30 I s1 t1-10 I (Sqrt[5]) s1 t1-15 I s2 t1+5 I (Sqrt[5]) s2 t1+30 I (t1^2)-10 I (Sqrt[5]) (t1^2)-30 I s1 (t1^2)+10 I (Sqrt[5]) s1 (t1^2)+15 I t2-5 I (Sqrt[5]) t2-15 I s2 t2+5 I (Sqrt[5]) s2 t2-30 I t1 t2+10 I (Sqrt[5]) t1 t2+30 I s2 t1 t2-10 I (Sqrt[5]) s2 t1 t2-5 (Sqrt[2]) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]) √(1/(-5+(Sqrt[5]))((s1^2) (-2-(-5+(Sqrt[5])) t1+(-5+(Sqrt[5])) (t1^2))+(-5+(Sqrt[5])) (((t1-t2))^2)+(-5+(Sqrt[5])) s2 (t1-t2) (-1+2 t2)+(s2^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2))+s1 (-(-5+(Sqrt[5])) (-1+2 t1) (t1-t2)+s2 (4+(-5+(Sqrt[5])) t2-(-5+(Sqrt[5])) t1 (-1+2 t2)))))+(Sqrt[10]) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]) √(1/(-5+(Sqrt[5]))((s1^2) (-2-(-5+(Sqrt[5])) t1+(-5+(Sqrt[5])) (t1^2))+(-5+(Sqrt[5])) (((t1-t2))^2)+(-5+(Sqrt[5])) s2 (t1-t2) (-1+2 t2)+(s2^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2))+s1 (-(-5+(Sqrt[5])) (-1+2 t1) (t1-t2)+s2 (4+(-5+(Sqrt[5])) t2-(-5+(Sqrt[5])) t1 (-1+2 t2)))))))/((Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]))])/((((2+(-5+(Sqrt[5])) t1-(-5+(Sqrt[5])) (t1^2)))^2) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)])));

ii1=Simplify[Limit[i1,s1->1]-Limit[i1,s1->0]]

ii1=(1/(240 (Sqrt[2])))(5+2 (Sqrt[5])) (-1+s2) (1/(((2+(-5+(Sqrt[5])) t1-(-5+(Sqrt[5])) (t1^2)))^2)(Sqrt[2]) (((-1+s2))^2) (Sqrt[(((((-1+s2))^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2)))/(-5+(Sqrt[5])))]) (-8-4 (-5+(Sqrt[5])) t2+(-5+7 (Sqrt[5])) (t2^2)+t1 (-4 (-5+(Sqrt[5]))+2 (-45+7 (Sqrt[5])) t2-20 (-3+(Sqrt[5])) (t2^2))+(t1^2) (-5+7 (Sqrt[5])-20 (-3+(Sqrt[5])) t2+20 (-3+(Sqrt[5])) (t2^2)))-1/(((-1+t1))^2)(Sqrt[2]) (Sqrt[(((-5+(Sqrt[5])) (((t1-t2))^2)+(-5+(Sqrt[5])) s2 (t1-t2) (-1+2 t2)+(s2^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2)))/(-5+(Sqrt[5])))]) (4 (((s2-t1+t2-s2 t2))^2)+(((-1+s2) (s2-t1+t2-s2 t2) (12-(11+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t1^2) (-7+6 t2)+t1 (-31+7 (Sqrt[5])-5 (-5+(Sqrt[5])) t2)))/(-2-(-5+(Sqrt[5])) t1+(-5+(Sqrt[5])) (t1^2)))+((((-1+s2))^2) (t1-t2) (-6+6 (Sqrt[5])-t2-9 (Sqrt[5]) t2+(t1^2) (-130+38 (Sqrt[5])+(125-55 (Sqrt[5])) t2)+(t1^3) (85-23 (Sqrt[5])+30 (-3+(Sqrt[5])) t2)+t1 (67-21 (Sqrt[5])+(-50+34 (Sqrt[5])) t2)))/(((2+(-5+(Sqrt[5])) t1-(-5+(Sqrt[5])) (t1^2)))^2))+(6 I (((-1+s2))^3) (((t1-t2))^2) (5+(Sqrt[5])-5 t2+5 t1 (-1+2 t2)) Log[(4 (-1+t1) (-10 I+2 I (Sqrt[5])+15 I t2-5 I (Sqrt[5]) t2+5 I (-3+(Sqrt[5])) t1 (-1+2 t2)-5 (Sqrt[2]) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]) (Sqrt[(((((-1+s2))^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2)))/(-5+(Sqrt[5])))])+(Sqrt[10]) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]) (Sqrt[(((((-1+s2))^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2)))/(-5+(Sqrt[5])))])-I s2 (2 (-5+(Sqrt[5]))-5 (-3+(Sqrt[5])) t2+5 (-3+(Sqrt[5])) t1 (-1+2 t2))))/((Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]))])/((((2+(-5+(Sqrt[5])) t1-(-5+(Sqrt[5])) (t1^2)))^2) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]))-(6 I (((-1+s2))^3) (((t1-t2))^2) (5+(Sqrt[5])-5 t2+5 t1 (-1+2 t2)) Log[(4 (-1+t1) (-15 I t1+5 I (Sqrt[5]) t1+30 I (t1^2)-10 I (Sqrt[5]) (t1^2)+15 I t2-5 I (Sqrt[5]) t2-30 I t1 t2+10 I (Sqrt[5]) t1 t2-I s2 (2 (-5+(Sqrt[5]))-5 (-3+(Sqrt[5])) t2+5 (-3+(Sqrt[5])) t1 (-1+2 t2))-5 (Sqrt[2]) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]) (Sqrt[(((-5+(Sqrt[5])) (((t1-t2))^2)+(-5+(Sqrt[5])) s2 (t1-t2) (-1+2 t2)+(s2^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2)))/(-5+(Sqrt[5])))])+(Sqrt[10]) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]) (Sqrt[(((-5+(Sqrt[5])) (((t1-t2))^2)+(-5+(Sqrt[5])) s2 (t1-t2) (-1+2 t2)+(s2^2) (-2-(-5+(Sqrt[5])) t2+(-5+(Sqrt[5])) (t2^2)))/(-5+(Sqrt[5])))])))/((Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)]))])/((((2+(-5+(Sqrt[5])) t1-(-5+(Sqrt[5])) (t1^2)))^2) (Sqrt[-5+(Sqrt[5])-5 (-3+(Sqrt[5])) t1+5 (-3+(Sqrt[5])) (t1^2)])));

NIntegrate[ii1,{t1,0,1},{s2,0,1},{t2,0,1},WorkingPrecision->10]

(iii1=FullSimplify[ii1,0<t1<1&&0<s2<1&&0<t2<1])//Timing

(i2=Integrate[iii1,t1])//Timing

Monte Carlo

ineq[x,y]/.a->1

f[x_,y_]:=(Sqrt[10]) (2+5 x)≥15 (Sqrt[2]) x+2 (Sqrt[25-5 (Sqrt[5])]) y&&2 (Sqrt[10])≥5 (Sqrt[2]) (-3+(Sqrt[5])) x+2 (Sqrt[25-5 (Sqrt[5])]) y&&2 (Sqrt[10])+5 (Sqrt[2]) (-1+(Sqrt[5])) x+2 (Sqrt[125-55 (Sqrt[5])]) y≥0&&(Sqrt[3+(Sqrt[5])])+2 (Sqrt[5-(Sqrt[5])]) y≥0&&2 (Sqrt[10])+2 (Sqrt[125-55 (Sqrt[5])]) y≥5 (Sqrt[2]) (-1+(Sqrt[5])) x

Length[pts=Select[Table[{RandomReal[{minx,maxx}],RandomReal[{miny,maxy}]},{10^6},{2}],And@@f@@@#&]]//Timing

(((((((1/4) (Sqrt[5 (5+2 (Sqrt[5]))])))/((maxx-minx)(maxy-miny)))))^2)//N

Through[{Min,Mean,Max}[ds=Norm/@(Subtract@@@pts)]]

Put[ds,"/tmp/data.m"]

Length[ds=Get["/tmp/data.m"]]

Histogram[ds]

Length[pts=Select[Table[{RandomReal[{minx,maxx}],RandomReal[{miny,maxy}]},{10^7},{2}],And@@f@@@#&]]//Timing

Through[{Min,Mean,Max}[ds=Norm/@(Subtract@@@pts)]]

FullSimplify[(2/((((1/4) (Sqrt[5 (5+2 (Sqrt[5]))])))^2))]

N[%]

Triangle Areas

TriangleArea[{{x1_,y1_},{x2_,y2_},{x3_,y3_}}]:=Abs[-x2 y1+x3 y1+x1 y2-x3 y2-x1 y3+x2 y3]/2

Monte Carlo

ineq[x,y]/.a->1

f[x_,y_]:=(Sqrt[10]) (2+5 x)≥15 (Sqrt[2]) x+2 (Sqrt[25-5 (Sqrt[5])]) y&&2 (Sqrt[10])≥5 (Sqrt[2]) (-3+(Sqrt[5])) x+2 (Sqrt[25-5 (Sqrt[5])]) y&&2 (Sqrt[10])+5 (Sqrt[2]) (-1+(Sqrt[5])) x+2 (Sqrt[125-55 (Sqrt[5])]) y≥0&&(Sqrt[3+(Sqrt[5])])+2 (Sqrt[5-(Sqrt[5])]) y≥0&&2 (Sqrt[10])+2 (Sqrt[125-55 (Sqrt[5])]) y≥5 (Sqrt[2]) (-1+(Sqrt[5])) x

Length[pts=Partition[Select[Table[{RandomReal[{minx,maxx}],RandomReal[{miny,maxy}]},{10^5}],f@@#&],3]]//Timing

Graphics[Transpose[{{Red,Green,Blue},Point/@Transpose[Take[pts,10^3]]}]]

Through[{Min,Mean,Max}[ds=TriangleArea/@pts]]

(1/180)(9+2(Sqrt[5]))(1/4)5Cot[π/5]//N

Histogram[ds]

Length[pts=Select[Table[{RandomReal[{minx,maxx}],RandomReal[{miny,maxy}]},{10^7},{2}],And@@f@@@#&]]//Timing

Through[{Min,Mean,Max}[ds=Norm/@(Subtract@@@pts)]]

FullSimplify[(2/((((1/4) (Sqrt[5 (5+2 (Sqrt[5]))])))^2))]

N[%]