Regular Pentagon
Author
Eric W. Weisstein
Title
Regular Pentagon
Description
The regular pentagon is the regular polygon with five sides, as illustrated above. A number of distance relationships between vertices of the regular pentagon can be derived by similar triangles in the above left figure, d/1=1/(1/phi)=phi, (1) where d is the diagonal distance. But the dashed vertical line connecting two nonadjacent polygon vertices is the same length as the diagonal one, so phi=1+1/phi (2) phi^2-phi-1. (3) Solving the quadratic equation and taking the plus sign...
Category
Educational Materials
Keywords
URL
http://www.notebookarchive.org/2019-07-0z5jtrw/
DOI
https://notebookarchive.org/2019-07-0z5jtrw
Date Added
2019-07-02
Date Last Modified
2019-07-02
File Size
0.68 megabytes
Supplements
Rights
Redistribution rights reserved
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Regular Pentagon
Regular Pentagon
Author
Author
Eric W. Weisstein
July 3, 2018
July 3, 2018
©2018 Wolfram Research, Inc. except for portions noted otherwise
Definitions
Definitions
ineq=Sqrt[10](2a+5x)≥15Sqrt[2]x+2Sqrt[25-5Sqrt[5]]y&&2Sqrt[10]a≥5Sqrt[2](-3+Sqrt[5])x+2Sqrt[25-5Sqrt[5]]y&&2Sqrt[10]a+5Sqrt[2](-1+Sqrt[5])x+2Sqrt[125-55Sqrt[5]]y≥0&&Sqrt[3+Sqrt[5]]a+2Sqrt[5-Sqrt[5]]y≥0&&2Sqrt[10]a+2Sqrt[125-55Sqrt[5]]y≥5Sqrt[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
Figure
In[]:=
Show[LaminaData["FilledRegularPentagon","Diagram"],ImageSize300]
Out[]=
Plots
Plots
Vertex Coordinates
Vertex Coordinates
Properties
Properties
Diagonal
Diagonal
Properties (point-up orientation)
Properties (point-up orientation)
Rehashing named triangle objects...
Vertices
Vertices
FullSimplify[Through[{MinValue,MaxValue}[{x,ineq[x,y]},{x,y}]],a>0]
-(1+(1+
1
4
5
)a,1
4
5
)aFullSimplify[Through[{MinValue,MaxValue}[{y,ineq[x,y]},{x,y}]],a>0]
-(5+a
1
2
1+
a,2
5
1
10
5
)Inequality
Inequality
Area
Area
Area
Area
Assuming[assum,FullSimplify[Area[reg]]]
Area::nmet:Unable to compute the area of region Polygon(1+(5+Power[2])a,(-1-,-,-
1
4
5
)a,aRoot[1+Times[2]+Times[2]&,3,0],0,1
10
1
4
5
)a,aRoot[1+Times[2]+Times[2]&,3,0],-a
2
1
2
1+2Power[2]
a,a
2
1
2
1+2Power[2]
a.
AreaPolygon(1++80&,3],0,(5+a,-(1++80&,3],-,-,-
1
4
5
)a,aRoot[1-202
#1
4
#1
1
10
5
)1
4
5
)a,aRoot[1-202
#1
4
#1
a
2
1
2
1+
a,2
5
a
2
1
2
1+
a2
5
Assuming[assum,FullSimplify[Area[reg/.a1]]]
1
4
5(5+2
5
)Assuming[assum,FullSimplify[Area[implreg]]]
1
4
5(5+2
5
)2
a
Integrate
Integrate
Assuming[a>0,Area[ineq[x,y],{x,y}]]//FullSimplify//Timing
12.2886,
1
4
5(5+2
5
)2
a
RegionMeasure
RegionMeasure
Assuming[assum,FullSimplify[RegionMeasure[reg]]]
RegionMeasure::nmet:Unable to compute the measure of region Polygon(1+(5+Power[2])a,(-1-,-,-
1
4
5
)a,aRoot[1+Times[2]+Times[2]&,3,0],0,1
10
1
4
5
)a,aRoot[1+Times[2]+Times[2]&,3,0],-a
2
1
2
1+2Power[2]
a,a
2
1
2
1+2Power[2]
a.
RegionMeasurePolygon(1++80&,3],0,(5+a,-(1++80&,3],-,-,-
1
4
5
)a,aRoot[1-202
#1
4
#1
1
10
5
)1
4
5
)a,aRoot[1-202
#1
4
#1
a
2
1
2
1+
a,2
5
a
2
1
2
1+
a2
5
Assuming[assum,FullSimplify[RegionMeasure[reg/.a1]]]
1
4
5(5+2
5
)Assuming[assum,FullSimplify[RegionMeasure[implreg]]]
5
4
1+
2
5
2
a
AreaInertiaTensor
AreaInertiaTensor
Centroid
Centroid
Integrate
Integrate
General::compat:Combinatorica Graph and Permutations functionality has been superseded by preloaded functionaliy. The package now being loaded may conflict with this. Please see the Compatibility Guide for details.
System`EdgeColor::shdw:Symbol EdgeColor appears in multiple contexts {System`,Combinatorica`}; definitions in context System` may shadow or be shadowed by other definitions.
Assuming[a>0,Centroid[ineq[x,y],{x,y}]]//FullSimplify//Timing
{77.1729,{0,0}}
RegionCentroid
RegionCentroid
Assuming[assum,FullSimplify[RegionCentroid[reg]]]
RegionCentroid::nmet:Unable to compute the centroid of region Polygon(1+(5+Power[2])a,(-1-,-,-
1
4
5
)a,aRoot[1+Times[2]+Times[2]&,3,0],0,1
10
1
4
5
)a,aRoot[1+Times[2]+Times[2]&,3,0],-a
2
1
2
1+2Power[2]
a,a
2
1
2
1+2Power[2]
a.
RegionCentroidPolygon(1++80&,3],0,(5+a,-(1++80&,3],-,-,-
1
4
5
)a,aRoot[1-202
#1
4
#1
1
10
5
)1
4
5
)a,aRoot[1-202
#1
4
#1
a
2
1
2
1+
a,2
5
a
2
1
2
1+
a2
5
Assuming[assum,FullSimplify[RegionCentroid[reg/.a1]]]
{0,0}
Assuming[assum,FullSimplify[RegionCentroid[implreg]]]
{0,0}
Convex
Convex
TimeConstrained[Region`ConvexRegionQ[reg],600]//Timing
TimeConstrained[Region`ConvexRegionQ[implreg],600]//Timing
Diagonals
Diagonals
EdgeLengths
EdgeLengths
GeneralizedDiameter
GeneralizedDiameter
Perimeter
Perimeter
RadiiOfGyration
RadiiOfGyration
Properties (point-right orientation)
Properties (point-right orientation)
Properties (arbitrary orientation)
Properties (arbitrary orientation)
Lengths
Lengths
Triangle Areas
Triangle Areas
Cite this as: Eric W. Weisstein, "Regular Pentagon" from the Notebook Archive (2018), https://notebookarchive.org/2019-07-0z5jtrw
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