Regular Decagon

Author

Eric W. Weisstein
July 3, 2018
This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polygons/RegularDecagon.nb.
For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/RegularDecagon.html.
©2018 Wolfram Research, Inc. except for portions noted otherwise

Definitions

ineq=2Sqrt[5+2Sqrt[5]]a+y≥Sqrt[2(5+Sqrt[5])]x+Sqrt[5]y&&(1+Sqrt[5])(Sqrt[5-2Sqrt[5]]x+y)≤2Sqrt[5+2Sqrt[5]]a&&Sqrt[5+2Sqrt[5]]a≥2y&&2Sqrt[5+2Sqrt[5]]a+(1+Sqrt[5])(Sqrt[5-2Sqrt[5]]x-y)≥0&&2Sqrt[5+2Sqrt[5]]a+Sqrt[2(5+Sqrt[5])]x+y≥Sqrt[5]y&&2Sqrt[5+2Sqrt[5]]a+Sqrt[2(5+Sqrt[5])]x+Sqrt[5]y≥y&&2Sqrt[5+2Sqrt[5]]a+Sqrt[10-2Sqrt[5]]x+y+Sqrt[5]y≥0&&Sqrt[5+2Sqrt[5]]a+2y≥0&&2Sqrt[5+2Sqrt[5]]a≥(1+Sqrt[5])(Sqrt[5-2Sqrt[5]]x-y)&&2Sqrt[5+2Sqrt[5]]a+Sqrt[5]y≥Sqrt[2(5+Sqrt[5])]x+y;
implreg=ImplicitRegion[ineq,{x,y}];
assum=a>0;
verts=a{{1/2(1+Sqrt[5]),0},{1/4(3+Sqrt[5]),Sqrt[5/8+Sqrt[5]/8]},{1/2,1/2Sqrt[5+2Sqrt[5]]},{-(1/2),1/2Sqrt[5+2Sqrt[5]]},{1/4(-3-Sqrt[5]),Sqrt[5/8+Sqrt[5]/8]},{1/2(-1-Sqrt[5]),0},{1/4(-3-Sqrt[5]),-(1/2)Sqrt[1/2(5+Sqrt[5])]},{-(1/2),-(1/2)Sqrt[5+2Sqrt[5]]},{1/2,-(1/2)Sqrt[5+2Sqrt[5]]},{1/4(3+Sqrt[5]),-(1/2)Sqrt[1/2(5+Sqrt[5])]}};
reg=Polygon[verts];

Figure

In[]:=
Show[LaminaData["FilledRegularDecagon","Diagram"],ImageSize300]
Out[]=

Plots

Diagram

LaminaData["FilledRegularDecagon","Diagram"]

DiscretizeRegion

Block[{a=1},DiscretizeRegion[#]]&/@{reg,implreg}
MinValue::ztest:Unable to decide whether numeric quantities 
2-4
2(5+Times[2])(5+Times[2])
+2
(5+Times[2])(5+Times[2])
-2
1
+4
1
+2
22(5+1)
Plus[2]
+2
10(5+Times[2])(5+Power[2])(5+Times[2])
Plus[2]
(1+
5
)-2+
2(5+Times[2])
Plus[2]
+
10(5+Times[2])
Plus[2]
-
5+Times[2]
+
5(5+Times[2])
-
5+Power[2]
-
5Plus[2]

,1,1,MaxAbs[1],Abs[1],4,Abs[5+1],Abs5+
1
1
1+1
 are equal to zero. Assuming they are.


,


Polygon


Properties

TableForm[FullSimplify[RegularPolygonInformation[10]],TableDepth2]
sides n
10
vertex angle α {rad, degrees}

4π
5
,
4π
5

central angle β {rad, degrees}

π
5
,36°
inradius r
1
2
5+2
5
circumradius R
1
2
(1+
5
)
area A
5
2
5+2
5

Equations


Properties

Rehashing named triangle objects...

Area

Area

Assuming[assum,FullSimplify[Area[reg/.a1]]]
5
2
5+2
5
Assuming[assum,FullSimplify[Area[implreg/.a1]]]
5
2
5+2
5

Integrate

Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing
7.5881,
5
2
5+2
5
2
a


RegionMeasure

Assuming[assum,FullSimplify[RegionMeasure[reg/.a1]]]
5
2
5+2
5
Assuming[assum,FullSimplify[RegionMeasure[implreg/.a1]]]
5
2
5+2
5

AreaInertiaTensor

Assuming[a>0,AreaInertiaTensor[ineq[x,y],{x,y}]]//FullSimplify//Timing
62.2486,
5
48
1025+458
5
4
a
,0,0,
5
48
1025+458
5
4
a


Centroid

Integrate

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

RegionCentroid

Convex

GeneralizedDiameter

RadiiOfGyration

Region

Lamina

Diagonals