Pentagonal Pyramid

Author

Eric W. Weisstein
February 27, 2007

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/PentagonalPyramid.nb.

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

©2007 Wolfram Research, Inc. except for portions noted otherwise

Solid

PolyhedronData["PentagonalPyramid","Orientations"]

PolyhedronData["PentagonalPyramid"]

Solid and Dual

PolyhedronData["PentagonalPyramid",#]&/@{"Graphics3D","Dual","DualCompound"}

Obsolete Pyramid package

Show[Graphics3D[Pyramid[5,1.4]],Boxed->False]

As a Johnson Solid

Show[Graphics3D[Polyhedron[JohnsonSolid[2]]],Boxed->False,ViewPoint->-{2,-2,2}]

PolyhedronName[JohnsonSolid[2]]

Wireframe

Show[Graphics[HiddenWireframe[Pyramid[5,1],BoxRatios->{1,1,1.4}]],AspectRatio->Automatic]

Net

PolyhedronData["PentagonalPyramid","Net"]

PolyhedronData["PentagonalPyramid","Net","Colored"]

NetPrintout[PolyhedronData["PentagonalPyramid","Polyhedron"],"PentagonalPyramid.pdf"]

Properties

<<MathWorld`PolyhedronOperations`

p=PolyhedronData[pname="PentagonalPyramid","Polyhedron"]

Centroid

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

PolyhedronData[pname,"Centroid"]

RegionCentroid[p]//FullSimplify

PolyhedronCentroid[p]//FullSimplify

Circumsphere

PolyhedronData[pname,"Circumsphere"]

FullSimplify[%]//ToRadicals

Circumsphere[p]//FullSimplify

(sphere=MyCircumsphere[p]//FullSimplify)//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Convex

ConvexPolyhedronQ[p]

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

DihedralAngles[p]//FullSimplify//Tally

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union//Quiet

e=FullSimplify[Sqrt[h^2+a^2 Csc[Pi/n]^2/4]/.n->5]

Unit

ToRadicals[FullSimplify[Solve[(Sqrt[(1/10) (5+(Sqrt[5])) (a^2)+(h^2)])==a,h],a>0]]

Simplify[PolyhedronEdgeLengths[Polyhedron[Pyramid[5]]]]//Union

Simplify[PolyhedronEdgeLengths[Polyhedron[Pyramid[5,(Sqrt[(1/10) (5-(Sqrt[5]))]) a,a]]],a>0]//Union

Faces

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

((MomentOfInertia[p,{0,0,0}])/(Volume[p]))//RootReduce//Timing

PolyhedronInertiaTensor[p]//Timing

Insphere

sphere=PolyhedronData[pname,"Insphere"]

Insphere[p]//FullSimplify

sphere=MyInsphere[p]//FullSimplify

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

MeanCylindricalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSquareCylindricalRadius

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[i]

MeanSphericalRadius

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSquareSphericalRadius

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//Timing

(i=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[i]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

Slant Height

General

s=FullSimplify[Sqrt[h^2+a^2 Cot[Pi/n]^2/4]/.n->5]

Unit

Simplify[s/.h->hh,a>0]

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

RootReduce[%]

SurfaceArea[p]//RootReduce//Timing

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce//Timing

General

FullSimplify[PolyhedronSurfaceArea[Polyhedron[Pyramid[5,h,a]]],h>0&&a>0]//Timing

Unit Edges

FullSimplify[PolyhedronSurfaceArea[Polyhedron[Pyramid[5,(Sqrt[(1/10) (5-(Sqrt[5]))]) a,a]]],a>0]//Timing

VertexSubsetHulls

PolyhedronData[pname,"VertexSubsetHulls"]

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p,Method->"GaussLaw"]//FullSimplify

General

FullSimplify[PolyhedronVolume[Polyhedron[Pyramid[5,h, a]]],h>0&&a>0]//Timing

With[{n=5},(1/12)n Cot[(π/n)](a^2)h]//FullSimplify

Unit Edges

FullSimplify[PolyhedronVolume[Polyhedron[Pyramid[5,(Sqrt[(1/10) (5-(Sqrt[5]))]) a,a]]],a>0]//Timing