Prism

Author

Eric W. Weisstein
April 26, 2022

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

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

©2022 Wolfram Research, Inc. except for portions noted otherwise

Graphics

PolyhedronData

PolyhedronData["Prism*"]

PolyhedronData/@%

Solid and Dual

GraphicsRow[Table[Graphics3D[{Opacity[.5],Red,PolyhedronData[{"Prism",n},"Polyhedron"],
Yellow,PolyhedronData[{"Prism",n},"Midsphere"],
Green,PolyhedronData[{"Prism",n},"Dual","Polyhedron"]},Boxed->False,Method->{"ShrinkWrap"->True}],{n,3,6}]]

Built-in

?Prism

Graphics3D[Prism[{{1,0,1},{0,0,0},{2,0,0},{1,2,1},{0,2,0},{2,2,0}}]]

Graphics3D[Prism[]]

Irregular

MyPrism::usage="MyPrism[{v1, ..., vn}, h, {x, y}] generates a prism having top and bottom of shape v, height h, and top-bottom offset of {x, y}.";

MyPrism[v_List,h_:1]:=MyPrism[v,h,{0,0}]
MyPrism[v_List,h_,off_List]:=Module[{i,n=Length[v],bot,top,sides,addn},
addn[n_,k_,m_]:=Mod[n+k-1,m]+1;
bot=Table[Append[v[[i]],0],{i,n}];
top=Table[Append[v[[i]]+off,h],{i,n}];
sides=Table[{bot[[i]],bot[[addn[i,1,n]]],top[[addn[i,1,n]]],top[[i]]},{i,n}];
Graphics3D[Flatten[{Polygon[bot],Polygon[top],Polygon/@sides}]]
]

With[
{a={{0,0},{.3,.1},{.5,.6},{.1,.8},{-.1,.6},{-.3,.8},{-.3,-.1},{-.04,-.2}}},
GraphicsRow[{MyPrism[a,1,{.5,0}],MyPrism[a,1,{0,0}]}]
]

Regular

Solids

From scratch

MyRegularPrism[n_Integer?Positive]:=MyRegularPrism[n,1,1]/;n>2
MyRegularPrism[n_Integer?Positive,h_:1]:=MyRegularPrism[n,h,1]/;n>2
MyRegularPrism[n_Integer?Positive,h_?Positive,s_?Positive]:=Module[
{i,z,R=s Csc[Pi/n]/2,verts,faces,r1=Range[n],r2=Range[n+1,2n]},
verts=Flatten[Table[{R Cos[2Pi i/n],R Sin[2Pi i/n],z h/2},{z,-1,1,2},{i,0,n-1}],1];
faces=Join[{Reverse[r1],r2},Flatten[{#1,Reverse[#2]}]&@@@Transpose[Partition[#,2,1,1]&/@{r1,r2}]];
Graphics3D[GraphicsComplex[verts,Polygon[faces]],Boxed->False]
]/;n>2

GraphicsGrid[Partition[Table[MyRegularPrism[n],{n,3,10}],UpTo[4]],Dividers->All]

PolyhedronData

GraphicsGrid[Partition[Table[Show[PolyhedronData[{"Prism",n}],Boxed->False],{n,3,10}],UpTo[4]],Dividers->All]

Nets

From scratch

MyPrismNet[n_,h_:1]:=MyPrismNet[n,h,1]
MyPrismNet[n_,h_,s_]:=Module[
{half=Floor[n/2],r=s Cot[Pi/n]/2,R=s Csc[Pi/n]/2,bot,top,ribbon,dy,i},
dy=If[OddQ[n],R,r];
bot=Table[{half s+s/2+R Sin[2Pi(i-3/2)/n],dy+R Cos[2Pi(i-3/2)/n]},{i,0,n}];
top=Table[{half s+s/2+R Sin[2Pi(i-3/2)/n],dy+h-R Cos[2Pi(i-3/2)/n]+2r},{i,0,n}];
ribbon=Table[{{s i,r+dy},{s(i+1),r+dy},{s(i+1),r+h+dy},{s i,r+h+dy},{s i,r+dy}},{i,0,n-1}];
Join[Line/@{bot,top},Line/@ribbon]
]

GraphicsGrid[Partition[Table[Graphics[MyPrismNet[n],AspectRatio->Automatic,PlotRange->All],{n,3,10}],UpTo[4]]]

PolyhedronData

GraphicsGrid[Partition[Table[PolyhedronData[{"Prism",n},"Net"],{n,3,10}],UpTo[4]],Dividers->All]

Properties

{"Prism", n}

Centroid

Table[PolyhedronData[{"Prism",n},"Centroid"],{n,3,10}]

Circumradius

Table[PolyhedronData[{"Prism",n},"Circumradius"],{n,3,10}]

Table[Sqrt[1+Csc[Pi/n]^2]/2,{n,3,10}]//FullSimplify

DualScale

Table[PolyhedronData[{"Prism",n},"DualScale"],{n,3,10}]

Table[Piecewise[{{Csc[(π/n)] Csc[((2 π)/n)],n==3}},Sec[(π/n)]],{n,3,10}]

Table[Graphics3D[{Opacity[.5],
Red,PolyhedronData[{"Prism",n},"Polyhedron"],
Yellow,PolyhedronData[{"Prism",n},"Midsphere"],
Green,PolyhedronData[{"CanonicalDipyramid",n},"Polyhedron"]/.Polyhedron[v_,f_]:>Polyhedron[Piecewise[{{Csc[(π/n)] Csc[((2 π)/n)],n==3}},Sec[(π/n)]]v,f]
}],{n,3,6}]

Table[Graphics3D[{Opacity[.5],
Red,PolyhedronData[{"Prism",n},"Polyhedron"],
Yellow,PolyhedronData[{"Prism",n},"Midsphere"],
Green,PolyhedronData[{"CanonicalDipyramid",n},"Polyhedron"]/.Polyhedron[v_,f_]:>Polyhedron[PolyhedronData[{"Prism",n},"DualScale"]v,f]
}],{n,3,6}]

Table[Graphics3D[{Opacity[.5],
Red,PolyhedronData[{"Prism",n},"Polyhedron"],
Yellow,PolyhedronData[{"Prism",n},"Midsphere"],
Green,PolyhedronData[{"Prism",n},"Dual","Polyhedron"]
}],{n,3,10}]

EdgeLengths

Table[PolyhedronData[{"Prism",n},"EdgeLengths"],{n,3,10}]

EdgeCount

Table[PolyhedronData[{"Prism",n},"EdgeCount"],{n,3,10}]

FindSequenceFunction[%,n-2]//Factor

FaceCount

Table[PolyhedronData[{"Prism",n},"FaceCount"],{n,3,10}]

FindSequenceFunction[%,n-2]//Factor

InertiaTensor

Table[PolyhedronData[{"Prism",n},"InertiaTensor"],{n,3,10}]

Inradius

Table[PolyhedronData[{"Prism",n},"Inradius"],{n,3,10}]

Midradius

Table[PolyhedronData[{"Prism",n},"Midradius"],{n,3,10}]

Table[(1/2)Csc[(Pi/n)],{n,3,10}]//FullSimplify

SurfaceArea

Table[PolyhedronData[{"Prism",n},"SurfaceArea"],{n,3,10}]

Table[n (1+1/2 Cot[Pi/n]),{n,3,10}]//FullSimplify

RootReduce[%]

VertexCount

Table[PolyhedronData[{"Prism",n},"VertexCount"],{n,3,10}]

FindSequenceFunction[%,n-2]//Factor

Volume

Table[PolyhedronData[{"Prism",n},"Volume"],{n,3,10}]

Table[n Cot[Pi/n]/4,{n,3,10}]//FullSimplify

RootReduce[%]

{"Prism", 3}

<<MathWorld`PolyhedronOperations`

Graphics3D[p=PolyhedronData[pname={"Prism",3},"Polyhedron"]]

Circumsphere

PolyhedronData[{"Prism",3},"Circumsphere"]

Norm/@PolyhedronData[{"Prism",3},"VertexCoordinates"]//FullSimplify//Union

Dual

PolyhedronData[{"Prism",3},"Dual",#]&/@{"Name","Scale"}

N[%]

With[{p={"Prism",3}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v1[[{2,4}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[5]],Mean[v1[[{2,4}]]]}],InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",3},"VertexCoordinates"][[All,-1]]//Max

FullSimplify[((2/3)/(Sqrt[(2/3)]))]

N[%]

Insphere

PolyhedronData[{"Prism",3},"Insphere"]

PolyhedronData[{"Prism",3},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

MeanCylindricalRadius

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

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]//FullSimplify

(i=((Integrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[%]

Midsphere

PolyhedronData[{"Prism",3},"Midsphere"]

PolyhedronData[{"Prism",3},"Lines"]/.Line[l_]:>Norm[Mean[l]]//Union

Graphics3D[{Opacity[.5],PolyhedronData[{"Prism",3},"Midsphere"],
{Red,PolyhedronData[{"Prism",3},"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[{"Prism",3},"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]

{"Prism", 4} == "Cube"

Circumsphere

PolyhedronData[{"Prism",4},"Circumsphere"]

Norm/@PolyhedronData[{"Prism",4},"VertexCoordinates"]//FullSimplify//Union

Dual

PolyhedronData[{"Prism",4},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",4}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

Insphere

PolyhedronData[{"Prism",4},"Insphere"]

PolyhedronData[{"Prism",4},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

Midsphere

PolyhedronData[{"Prism",4},"Midsphere"]

PolyhedronData[{"Prism",4},"Lines"]/.Line[l_]:>Norm[Mean[l]]//FullSimplify//Union

With[{p={"Prism",4}},Graphics3D[{Opacity[.5],Red,PolyhedronData[p,"Midsphere"],
{Yellow,PolyhedronData[p,"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[p,"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]]

{"Prism", 5}

Initialization

<<MathWorld`Polyhedra`

Graphics3D[p=PolyhedronData[pname={"Prism",5},"Polyhedron"]]

vol=PolyhedronData[pname,"Volume"]

Circumsphere

PolyhedronData[{"Prism",5},"Circumsphere"]

N[%]

Norm/@PolyhedronData[{"Prism",5},"VertexCoordinates"]//FullSimplify//Union

N[%]

Dual

PolyhedronData[{"Prism",5},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",5}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v2[[{3,5}]]//Mean//FullSimplify

v1[[{4,3}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}],InfiniteLine[{v2[[6]],Mean[v1[[{2,10}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",5},"VertexCoordinates"][[All,-1]]//Max

FullSimplify[((1+(1/(Sqrt[5])))/(Sqrt[(1/10) (5-(Sqrt[5]))]))]

Insphere

PolyhedronData[{"Prism",5},"Insphere"]

PolyhedronData[{"Prism",5},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

MeanCylindricalRadius [unevaluated integral]

PolyhedronData[pname,"MeanCylindricalRadius"]

N[%]

Integrate over polyhedron

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/vol)

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

Divergence theorem

(rxy=MeanCylindricalRadius[p,"Simplification"->FullSimplify,Debug->True])//Timing

N[rxy]

ResourceFunction["LogSimplify"][rxy]//FullSimplify

Midsphere

PolyhedronData[{"Prism",5},"Midsphere"]

PolyhedronData[{"Prism",5},"Lines"]/.Line[l_]:>Norm[Mean[l]]//FullSimplify//Union

Graphics3D[{Opacity[.5],PolyhedronData[{"Prism",5},"Midsphere"],
{Red,PolyhedronData[{"Prism",5},"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[{"Prism",5},"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]

{"Prism", 6}

<<MathWorld`PolyhedronOperations`

Graphics3D[p=PolyhedronData[pname={"Prism",6},"Polyhedron"]]

Circumsphere

PolyhedronData[{"Prism",6},"Circumsphere"]

Norm/@PolyhedronData[{"Prism",6},"VertexCoordinates"]//FullSimplify//Union

Dual

PolyhedronData[{"Prism",6},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",6}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v1[[{2,4}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[5]],Mean[v1[[{2,4}]]]}],InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",3},"VertexCoordinates"][[All,-1]]//Max

Insphere

PolyhedronData[{"Prism",6},"Insphere"]

PolyhedronData[{"Prism",6},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

MeanCylindricalRadius

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

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]//FullSimplify

(i=((Integrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(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]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]//FullSimplify

((8NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{x,0,∞},{y,0,∞},{z,0,∞}])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

(ii=FullSimplify[i])//Timing

(iii=TrigToExp[ii]//FullSimplify)//Timing

<<Utilities`Simplify`

newi=LogSimplify[TrigToExp[ii]]

N[newi]

exp=(({(I/(72 (Sqrt[3]))),((11 I)/(72 (Sqrt[3]))),(I/(48 (Sqrt[3]))),(I/(32 (Sqrt[3]))),((I (Sqrt[3]))/64),(I/(288 (Sqrt[3])))})/(I/(576 (Sqrt[3]))))

ce=ComplexExpand[{(((((1-I (Sqrt[(3/5)]))/(1+I (Sqrt[(3/5)])))))^(-1)), (((((1-(I/(Sqrt[15])))/(1+(I/(Sqrt[15]))))))^(-1)) ,(((1-((I (Sqrt[15]))/7))/(1+((I (Sqrt[15]))/7)))), (((1-((7 I (Sqrt[15]))/17))/(1+((7 I (Sqrt[15]))/17)))), (((1-((119 I (Sqrt[15]))/223))/(1+((119 I (Sqrt[15]))/223)))) ,(((1-((33 I (Sqrt[15]))/7))/(1+((33 I (Sqrt[15]))/7))))}]//FullSimplify

exp

logexp=Times@@((ce^((I/(576 (Sqrt[3])))exp)))

FullSimplify[(2^(-((61 I)/(96 (Sqrt[3])))-((23 I (Sqrt[3]))/64)))]

N[((Sqrt[5])/8)-((29 π)/(288 (Sqrt[3])))+(7/4) Log[3-(Sqrt[5])]+(17/6) Log[(1/2)+((Sqrt[5])/2)]-(7/4) Log[-1+(Sqrt[5])]+Log[logexp],10]

N[((Sqrt[5])/8)-((29 π)/(288 (Sqrt[3])))+(7/4) Log[3-(Sqrt[5])]+(17/6) Log[(1/2)+((Sqrt[5])/2)]-(7/4) Log[-1+(Sqrt[5])]+Log[(((1+I (Sqrt[15])))^(I/(72 (Sqrt[3])))) (((7+I (Sqrt[15])))^((11 I)/(72 (Sqrt[3])))) (((17-7 I (Sqrt[15])))^(I/(48 (Sqrt[3])))) (((-223-119 I (Sqrt[15])))^(I/(32 (Sqrt[3])))) (((-8143-231 I (Sqrt[15])))^(I/(288 (Sqrt[3])))) (((-81343-26537 I (Sqrt[15])))^((I (Sqrt[3]))/64))]-((329 I)/(192 (Sqrt[3])))Log[2],10]

N[Log[(((1+I (Sqrt[15])))^(I/(72 (Sqrt[3])))) (((7+I (Sqrt[15])))^((11 I)/(72 (Sqrt[3])))) (((17-7 I (Sqrt[15])))^(I/(48 (Sqrt[3])))) (((-223-119 I (Sqrt[15])))^(I/(32 (Sqrt[3])))) (((-8143-231 I (Sqrt[15])))^(I/(288 (Sqrt[3])))) (((-81343-26537 I (Sqrt[15])))^((I (Sqrt[3]))/64))]]

I N[Log[(((1+I (Sqrt[15])))^(1/(72 (Sqrt[3])))) (((7+I (Sqrt[15])))^(11/(72 (Sqrt[3])))) (((17-7 I (Sqrt[15])))^(1/(48 (Sqrt[3])))) (((-223-119 I (Sqrt[15])))^(1/(32 (Sqrt[3])))) (((-8143-231 I (Sqrt[15])))^(1/(288 (Sqrt[3])))) (((-81343-26537 I (Sqrt[15])))^((Sqrt[3])/64))],20]

Table[(((1-I(Sqrt[15])))^n),{n,10}]//Expand

I N[Log[((((((1+I (Sqrt[15])))^(288/(72 (Sqrt[3])))) (((7+I (Sqrt[15])))^((288 11)/(72 (Sqrt[3])))) (((17-7 I (Sqrt[15])))^(288/(48 (Sqrt[3])))) (((-223-119 I (Sqrt[15])))^(288/(32 (Sqrt[3])))) (((-8143-231 I (Sqrt[15])))^(288/(288 (Sqrt[3])))) (((-81343-26537 I (Sqrt[15])))^((288(Sqrt[3]))/64))))^(1/288))],20]

288/64

N[%]

N[((Sqrt[5])/8)-((29 π)/(288 (Sqrt[3])))+(7/4) Log[3-(Sqrt[5])]+(17/6) (Log[1+(Sqrt[5])]+Log[1/2])-(7/4) Log[-1+(Sqrt[5])]+Log[logexp],10]

Expand[(1/2)(((1-(Sqrt[5])))^2)]//Factor

Log[(((1-(Sqrt[5])))^2)]//N

-2Pi I+2Log[(1-(Sqrt[5]))]//N

N[((Sqrt[5])/8)-((29 π)/(288 (Sqrt[3])))+(7/4) (Log[1/2]+2Log[1-(Sqrt[5])]-2Pi I)+(17/6) (Log[1+(Sqrt[5])]+Log[1/2])-(7/4) Log[-1+(Sqrt[5])]+Log[logexp],10]

Expand[((Sqrt[5])/8)-((29 π)/(288 (Sqrt[3])))+(7/4) (Log[1/2]+2Log[1-(Sqrt[5])]-2Pi I)+(17/6) (Log[1+(Sqrt[5])]+Log[1/2])-(7/4) Log[-1+(Sqrt[5])]]

N[((Sqrt[5])/8)-((29 π)/(288 (Sqrt[3])))-((55 Log[2])/12)+(7/4) Log[-1+(Sqrt[5])]+(17/6) Log[1+(Sqrt[5])]+Log[logexp],10]

Log[(((((1-I (Sqrt[(3/5)]))/(1+I (Sqrt[(3/5)])))))^(-(I/(72 (Sqrt[3]))))) (((((1-(I/(Sqrt[15])))/(1+(I/(Sqrt[15]))))))^(-((11 I)/(72 (Sqrt[3]))))) (((((1-((I (Sqrt[15]))/7))/(1+((I (Sqrt[15]))/7)))))^(I/(48 (Sqrt[3])))) (((((1-((7 I (Sqrt[15]))/17))/(1+((7 I (Sqrt[15]))/17)))))^(I/(32 (Sqrt[3])))) (((((1-((119 I (Sqrt[15]))/223))/(1+((119 I (Sqrt[15]))/223)))))^((I (Sqrt[3]))/64)) (((((1-((33 I (Sqrt[15]))/7))/(1+((33 I (Sqrt[15]))/7)))))^(I/(288 (Sqrt[3]))))]//N

logexp//N

log=(((Times@@((ce^exp))))^(I/(576 (Sqrt[3]))))//N

Times@@N[(ce^(exp (I/(576 (Sqrt[3])))))]

Log[(((((1-I (Sqrt[(3/5)]))/(1+I (Sqrt[(3/5)])))))^(-(I/(72 (Sqrt[3]))))) (((((1-(I/(Sqrt[15])))/(1+(I/(Sqrt[15]))))))^(-((11 I)/(72 (Sqrt[3]))))) (((((1-((I (Sqrt[15]))/7))/(1+((I (Sqrt[15]))/7)))))^(I/(48 (Sqrt[3])))) (((((1-((7 I (Sqrt[15]))/17))/(1+((7 I (Sqrt[15]))/17)))))^(I/(32 (Sqrt[3])))) (((((1-((119 I (Sqrt[15]))/223))/(1+((119 I (Sqrt[15]))/223)))))^((I (Sqrt[3]))/64)) (((((1-((33 I (Sqrt[15]))/7))/(1+((33 I (Sqrt[15]))/7)))))^(I/(288 (Sqrt[3]))))]//N

MinValue[{#,ineq},{x,y,z}]&/@{x,y,z}//FullSimplify

MaxValue[{#,ineq},{x,y,z}]&/@{x,y,z}//FullSimplify

(i1=Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{z,0,1/2}])//Timing

((8NIntegrate[i1,{x,0,1},{y,0,((Sqrt[3])/2)}])/(PolyhedronData[pname,"Volume"]))

i11=FullSimplify[i1,0<x<1&&0<y<((Sqrt[3])/2)]

(i2=Assuming[0<x<1,FullSimplify@Integrate[i11,{y,0,((Sqrt[3])/2)}]])//Timing

((8NIntegrate[i2,{x,0,1}])/(PolyhedronData[pname,"Volume"]))

(i3=Integrate[i2,{x,0,1}])//Timing

N[((8i3)/(PolyhedronData[pname,"Volume"]))]

(i=FullSimplify[((8i3)/(PolyhedronData[pname,"Volume"]))])//Timing

N[i]

TrigToExp[i]//FullSimplify

(i2=Assuming[0<y<((Sqrt[3])/2),FullSimplify@Integrate[i11,{x,0,1}]])//Timing

((8NIntegrate[i2,{y,0,((Sqrt[3])/2)}])/(PolyhedronData[pname,"Volume"]))

(i3=Integrate[i2,{y,0,((Sqrt[3])/2)}])//Timing

N[((8i3)/(PolyhedronData[pname,"Volume"])),20]

N[i3]

Midsphere

PolyhedronData[{"Prism",6},"Midsphere"]

PolyhedronData[{"Prism",6},"Lines"]/.Line[l_]:>Norm[Mean[l]]//Union

With[{p={"Prism",6}},Graphics3D[{Opacity[.5],PolyhedronData[p,"Midsphere"],
{Red,PolyhedronData[p,"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[p,"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]]

{"Prism", 7}

Circumsphere

PolyhedronData[{"Prism",7},"Circumsphere"]

Norm/@PolyhedronData[{"Prism",7},"VertexCoordinates"]//FullSimplify//RootReduce//Union

Dual

PolyhedronData[{"Prism",7},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",7}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v1[[{2,4}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[5]],Mean[v1[[{2,4}]]]}],InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",3},"VertexCoordinates"][[All,-1]]//Max

Insphere

PolyhedronData[{"Prism",7},"Insphere"]

PolyhedronData[{"Prism",7},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//N//Union

Midsphere

PolyhedronData[{"Prism",7},"Midsphere"]

PolyhedronData[{"Prism",7},"Lines"]/.Line[l_]:>Norm[Mean[l]]//FullSimplify//Union

With[{p={"Prism",6}},Graphics3D[{Opacity[.5],PolyhedronData[p,"Midsphere"],
{Red,PolyhedronData[p,"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[p,"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]]

{"Prism", 8}

<<MathWorld`PolyhedronOperations`

Graphics3D[p=PolyhedronData[pname={"Prism",8},"Polyhedron"]]

Circumsphere

PolyhedronData[{"Prism",8},"Circumsphere"]

Norm/@PolyhedronData[{"Prism",8},"VertexCoordinates"]//FullSimplify//Union

Dual

PolyhedronData[{"Prism",8},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",8}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v1[[{2,4}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[5]],Mean[v1[[{2,4}]]]}],InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",3},"VertexCoordinates"][[All,-1]]//Max

Insphere

PolyhedronData[{"Prism",8},"Insphere"]

PolyhedronData[{"Prism",8},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

Midsphere

PolyhedronData[{"Prism",8},"Midsphere"]

PolyhedronData[{"Prism",8},"Lines"]/.Line[l_]:>Norm[Mean[l]]//FullSimplify//Union

N[%]

With[{p={"Prism",6}},Graphics3D[{Opacity[.5],PolyhedronData[p,"Midsphere"],
{Red,PolyhedronData[p,"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[p,"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]]

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]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]//FunctionExpand//FullSimplify

MinValue[{#,ineq},{x,y,z}]&/@{x,y,z}//FullSimplify

MaxValue[{#,ineq},{x,y,z}]&/@{x,y,z}//FullSimplify

(i=((8NIntegrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{x,0,∞},{y,0,∞},{z,0,∞}])/(PolyhedronData[pname,"Volume"])))//Timing

(i=((8Integrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{x,0,∞},{y,0,∞},{z,0,∞}])/(PolyhedronData[pname,"Volume"])))//Timing

N[i,20]

(i1=FullSimplify@Integrate[(Sqrt[(x^2)+(y^2)])Boole[ineq],{z,0,∞}])//Timing

((8NIntegrate[i1,{x,0,(Sqrt[1+(1/(Sqrt[2]))])},{y,0,(Sqrt[1+(1/(Sqrt[2]))])}])/(PolyhedronData[pname,"Volume"]))

(i2=FullSimplify@Integrate[i1,{y,0,(Sqrt[1+(1/(Sqrt[2]))])}])//Timing

((8NIntegrate[i2,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])/(PolyhedronData[pname,"Volume"]))

(i22=FullSimplify[i2,0<x<(Sqrt[1+(1/(Sqrt[2]))])])//Timing

((8NIntegrate[i22,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])/(PolyhedronData[pname,"Volume"]))//Timing

(i3=Integrate[i22,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])//Timing

N[((8i3)/(PolyhedronData[pname,"Volume"])),20]

(i=FullSimplify[((8i3)/(PolyhedronData[pname,"Volume"]))])//Timing

N[i]

MeanSphericalRadius

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

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]//FullSimplify

MinValue[{#,ineq},{x,y,z}]&/@{x,y,z}//FullSimplify

MaxValue[{#,ineq},{x,y,z}]&/@{x,y,z}//FullSimplify

((8NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{x,0,∞},{y,0,∞},{z,0,∞}])/(PolyhedronData[pname,"Volume"]))//Timing

(i1=FullSimplify@Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[ineq],{z,0,∞}])//Timing

((8NIntegrate[i1,{x,0,(Sqrt[1+(1/(Sqrt[2]))])},{y,0,(Sqrt[1+(1/(Sqrt[2]))])}])/(PolyhedronData[pname,"Volume"]))

(i11=FullSimplify[i1,0<x<(Sqrt[1+(1/(Sqrt[2]))])&&0<y<(Sqrt[1+(1/(Sqrt[2]))])])//Timing

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

Polar coordinates

reg=Region[ImplicitRegion[((2 x>(Sqrt[2+(Sqrt[2])])&&x+(Sqrt[2]) x+y≤(Sqrt[5+(7/(Sqrt[2]))]))||(2 x≤(Sqrt[2+(Sqrt[2])])&&(Sqrt[2]) x+y≤(Sqrt[1+(1/(Sqrt[2]))])+x))&&0<x<(Sqrt[1+(1/(Sqrt[2]))])&&0<y<(Sqrt[1+(1/(Sqrt[2]))]),{x,y}],Axes->True]

(Sqrt[1+(1/(Sqrt[2]))])-((Sqrt[2+(Sqrt[2])])/2)//FullSimplify

Solve[{x+(Sqrt[2]) x+y==(Sqrt[5+(7/(Sqrt[2]))]),(Sqrt[2]) x+y==(Sqrt[1+(1/(Sqrt[2]))])+x},{x,y}]//FullSimplify

Show[{reg,Graphics[Point[{{((Sqrt[2+(Sqrt[2])])/2),((Sqrt[2+(Sqrt[2])])/2)},{0,(Sqrt[1+(1/(Sqrt[2]))])},{(Sqrt[1+(1/(Sqrt[2]))]),0}}]]}]

eqn=Det[{{x,y,1},{(Sqrt[1+(1/(Sqrt[2]))]),0,1},{((Sqrt[2+(Sqrt[2])])/2),((Sqrt[2+(Sqrt[2])])/2),1}}]//FullSimplify

eqn/.{x->r Cos[θ],y->r Sin[θ]}//FullSimplify

Solve[%==0,r]//FullSimplify

FullSimplify[{(Sqrt[2+(Sqrt[2])]),(Sqrt[2-(Sqrt[2])])}/(1+(Sqrt[2]))]

(1/(((Sqrt[2+(Sqrt[2])])/(1+(Sqrt[2]))) Cos[θ]+((Sqrt[2-(Sqrt[2])])/(1+(Sqrt[2]))) Sin[θ]))//FullSimplify

((1+(Sqrt[2]))/((Sqrt[2+(Sqrt[2])]) Cos[θ]+(Sqrt[2-(Sqrt[2])]) Sin[θ]))-(1/((Sqrt[2-(Sqrt[2])]) Cos[θ]+(Sqrt[10-7 (Sqrt[2])]) Sin[θ]))//FullSimplify

PolarPlot[((1+(Sqrt[2]))/((Sqrt[2+(Sqrt[2])]) Cos[θ]+(Sqrt[2-(Sqrt[2])]) Sin[θ])),{θ,0,π/4},Epilog->Point[{{(Sqrt[1+(1/(Sqrt[2]))]),0},{((Sqrt[2+(Sqrt[2])])/2),((Sqrt[2+(Sqrt[2])])/2)}}]]

((16NIntegrate[(1/8) ((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r ,{θ,0,π/4},{r,0,((1+(Sqrt[2]))/((Sqrt[2+(Sqrt[2])]) Cos[θ]+(Sqrt[2-(Sqrt[2])]) Sin[θ]))}])/(PolyhedronData[pname,"Volume"]))

c==(Sqrt[(a^2)+(b^2)])/.{a->(Sqrt[2+(Sqrt[2])]),b->(Sqrt[2-(Sqrt[2])])}//FullSimplify

δ=ArcTan[-b/a]/.{a->(Sqrt[2+(Sqrt[2])]),b->(Sqrt[2-(Sqrt[2])])}//FullSimplify

((16NIntegrate[(1/8) ((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r ,{θ,0,π/4},{r,0,((1+(Sqrt[2]))/(2 Cos[θ-(π/8)]))}])/(PolyhedronData[pname,"Volume"]))

((16NIntegrate[(1/8) ((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r ,{θ,-π/8,π/8},{r,0,((1+(Sqrt[2]))/(2 Cos[θ]))}])/(PolyhedronData[pname,"Volume"]))

((32NIntegrate[(1/8) ((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r ,{θ,0,π/8},{r,0,((1+(Sqrt[2]))/(2 Cos[θ]))}])/(PolyhedronData[pname,"Volume"]))

FullSimplify[((1+(Sqrt[2]))/(2 Cos[θ]))]

((4NIntegrate[((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r ,{θ,0,π/8},{r,0,(1/2) (1+(Sqrt[2])) Sec[θ]}])/(PolyhedronData[pname,"Volume"]))

(i=((4Integrate[ ((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r ,{θ,0,π/8},{r,0,(1/2) (1+(Sqrt[2])) Sec[θ]}])/(PolyhedronData[pname,"Volume"])))//Timing

(res=Assuming[0<θ<π/8,Integrate[((Sqrt[1+4 (r^2)])+4 (r^2) ArcCsch[2 r])r,{r,0,(1/2) (1+(Sqrt[2])) Sec[θ]}]])//Timing

Plot[res,{θ,0,Pi/8}]

Integrate[res,θ]

((4NIntegrate[res,{θ,0,π/8}])/(PolyhedronData[pname,"Volume"]))

res2=FullSimplify[res,0<θ<π/8]

Plot[res2,{θ,-Pi/8,Pi/8}]

(i1=Integrate[(-4+2 (Sqrt[2]) (Sqrt[7+4 (Sqrt[2])+Cos[2 θ]]) Sec[θ]),{θ,0,Pi/8}]//FullSimplify)//Timing

(i2=Integrate[5 (4+3 (Sqrt[2])) (Sqrt[7+4 (Sqrt[2])+Cos[2 θ]]) ((Sec[θ])^3),{θ,0,Pi/8}]//FullSimplify)//Timing

(i3=Integrate[-6 (17+12 (Sqrt[2])) ArcSinh[1]((Sec[θ])^4),{θ,0,Pi/8}]//FullSimplify)//Timing

(i4=Integrate[-6 (17+12 (Sqrt[2])) (-Log[Cos[θ]+((Sqrt[7+4 (Sqrt[2])+Cos[2 θ]])/(Sqrt[2]))]) ((Sec[θ])^4),{θ,0,Pi/8}]//FullSimplify)//Timing

N[((4(i1+i2+i3+i4))/(96PolyhedronData[pname,"Volume"])),20]

FullSimplify[((4(i1+i2+i3+i4))/(96PolyhedronData[pname,"Volume"]))]

N[%,20]

FunctionExpand[ArcSinh[(Sqrt[2 (2+(Sqrt[2]))]) ((Sin[(π/8)])^2)]]//FullSimplify

Log[((NumericalApproximation[Root, (RowDefault["0.832", FEExpressionsNumericalApproximationElider])+(RowDefault["0.555", FEExpressionsNumericalApproximationElider]) I, Root[16-8 #1+((#1)^2)-8 ((#1)^3)+16 ((#1)^4)&,4], ])^(NumericalApproximation[Root, (RowDefault["16.3", FEExpressionsNumericalApproximationElider]) I, Root[17+266 ((#1)^2)+((#1)^4)&,2], ]))] //ToRadicals//FullSimplify//ToRadicals

i=(1/48) (-1+(Sqrt[2])) (-(π/2)-4 (11+8 (Sqrt[2])) ArcSinh[1]+5 ((Sqrt[23+16 (Sqrt[2])])+(8+6 (Sqrt[2])) ArcCsch[(Sqrt[2 (2+(Sqrt[2]))])])+4 (ArcCsc[2 (2+(Sqrt[2]))]+(1+(Sqrt[2])) ArcTanh[(Sqrt[(1/17) (5-2 (Sqrt[2]))])])+(1/(Sqrt[133+94 (Sqrt[2])]))(17+12 (Sqrt[2])) (-1+3 (Sqrt[2])+(Sqrt[46+32 (Sqrt[2])]) Log[2-(1/(Sqrt[2]))+(Sqrt[(7/2)-2 (Sqrt[2])])]-(Sqrt[61+38 (Sqrt[2])]) (Log[4]-2 Log[8+5 (Sqrt[2])+(Sqrt[46+32 (Sqrt[2])])])+-(Sqrt[133+94 (Sqrt[2])]) ArcCot[(Sqrt[(1151/833)+((512 (Sqrt[2]))/833)])]));

FullSimplify[i]

N[i,20]

((4NIntegrate[res2,{θ,0,π/8}])/(PolyhedronData[pname,"Volume"]))

((2NIntegrate[res2,{θ,-Pi/8,π/8}])/(PolyhedronData[pname,"Volume"]))

(i=Integrate[res2,{θ,-Pi/8,π/8}])//Timing

N[((2i)/(PolyhedronData[pname,"Volume"])),20]

FullSimplify[((2i)/(PolyhedronData[pname,"Volume"]))]//Timing

Integrate[res2,θ]

(i=Integrate[res2,{θ,0,π/8}])//Timing

((4N[i,20])/(PolyhedronData[pname,"Volume"]))

FullSimplify[i]

Cartesian

ii=FullSimplify[Integrate[((Sqrt[1+4 (x^2)+4 (y^2)])+4 ((x^2)+(y^2)) ArcCsch[2 (Sqrt[(x^2)+(y^2)])]),x]]

(i2=Assuming[0<x<(Sqrt[1+(1/(Sqrt[2]))]),Integrate[i11,{y,0,(Sqrt[1+(1/(Sqrt[2]))])}]])//Timing

((8NIntegrate[i2,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])/(PolyhedronData[pname,"Volume"]))

(i3=Integrate[i2,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])//Timing

(i22=FullSimplify[i2,0<x<(Sqrt[1+(1/(Sqrt[2]))])])//Timing

((8NIntegrate[i22,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])/(PolyhedronData[pname,"Volume"]))//Timing

(i3=Integrate[i22,{x,0,(Sqrt[1+(1/(Sqrt[2]))])}])//Timing

N[((8i3)/(PolyhedronData[pname,"Volume"])),20]

(i=FullSimplify[((8i3)/(PolyhedronData[pname,"Volume"]))])//Timing

N[i]

{"Prism", 9}

Circumsphere

PolyhedronData[{"Prism",9},"Circumsphere"]

Norm/@PolyhedronData[{"Prism",9},"VertexCoordinates"]//FullSimplify//Union

N[%]

Dual

PolyhedronData[{"Prism",9},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",9}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v1[[{2,4}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[5]],Mean[v1[[{2,4}]]]}],InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",3},"VertexCoordinates"][[All,-1]]//Max

Insphere

PolyhedronData[{"Prism",9},"Insphere"]

PolyhedronData[{"Prism",9},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

Midsphere

PolyhedronData[{"Prism",9},"Midsphere"]

PolyhedronData[{"Prism",9},"Lines"]/.Line[l_]:>Norm[Mean[l]]//FullSimplify//Union

With[{p={"Prism",6}},Graphics3D[{Opacity[.5],PolyhedronData[p,"Midsphere"],
{Red,PolyhedronData[p,"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[p,"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]]

{"Prism", 10}

Circumsphere

PolyhedronData[{"Prism",10},"Circumsphere"]

Circumsphere[PolyhedronData[{"Prism",10},"Polyhedron"]]//FullSimplify

Norm/@PolyhedronData[{"Prism",10},"VertexCoordinates"]//FullSimplify//Union

Dual

PolyhedronData[{"Prism",10},"Dual",#]&/@{"Name","Scale"}

With[{p={"Prism",10}},{
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v1=PolyhedronData[p,"VertexCoordinates"]]},
Red,PolyhedronData[p,"Polyhedron"]},Axes->True],
Graphics3D[{{Opacity[1],MapIndexed[Text[#2[[1]],#,Background->White]&,v2=PolyhedronData[p,"Dual","Polyhedron"][[1]]]},Green,PolyhedronData[p,"Dual","Polyhedron"]},Axes->True]
}]

both=Graphics3D[{Opacity[.3],First/@%},Boxed->False]

pt=v1[[{2,4}]]//Mean//FullSimplify

RegionIntersection[InfiniteLine[{v2[[5]],Mean[v1[[{2,4}]]]}],InfiniteLine[{v2[[3]],Mean[v1[[{4,6}]]]}]]//FullSimplify

PolyhedronData[{"Dipyramid",3},"VertexCoordinates"][[All,-1]]//Max

Insphere

PolyhedronData[{"Prism",10},"Insphere"]

Insphere[PolyhedronData[{"Prism",10},"Polyhedron"]]

PolyhedronData[{"Prism",10},"Polygons"]/.Polygon[l_]:>Norm[Mean[l]]//FullSimplify//Union

Midsphere

PolyhedronData[{"Prism",10},"Midsphere"]

PolyhedronData[{"Prism",10},"Lines"]/.Line[l_]:>Norm[Mean[l]]//FullSimplify//Union

With[{p={"Prism",10}},Graphics3D[{Opacity[.5],PolyhedronData[p,"Midsphere"],
{Red,PolyhedronData[p,"Dual","GraphicsComplex"]},
{Blue,PolyhedronData[p,"GraphicsComplex"]}},Boxed->False,Method->{"ShrinkWrap"->True}]]

Properties

MeanCylindricalRadius

Table[((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"])),{n,3,10}]

Table[((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"])),{n,3,3}]

N[%]

MeanCylindricalSquareRadius

Table[((NIntegrate[(x^2)+(y^2),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"])),{n,3,10}]

data=Table[{n,Sequence@@Timing[((Integrate[(x^2)+(y^2),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"]))//RootReduce//FullSimplify]},{n,3,10}];//Timing

TextGrid[data,Dividers->All]

MeanSphericalRadius

Table[((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"])),{n,3,10}]

Table[((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"])),{n,3,3}]

N[%]

MeanSphericalSquareRadius

Table[((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"])),{n,3,10}]

data=Table[{n,Sequence@@Timing[((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈PolyhedronData[{"Prism",n},"Polyhedron"]])/(PolyhedronData[{"Prism",n},"Volume"]))//RootReduce//FullSimplify]},{n,3,10}];//Timing

TextGrid[data,Dividers->All]

Construction

ConvexHull

Prism2[n_,h_,s_]:=ConvexHullRegion[CirclePoints[s Csc[Pi/n]/2,n,# h/2]&/@{-1,1}]
Prism2[n_,h_]:=

Prism2

Canonical

PolyhedronData[{"Prism",5},"Canonical"]

<<MathWorld`Polyhedra`

Graphics3D[FindCanonicalPolyhedron[PolyhedronData[{"Prism",6},"Polyhedron"]]]

PolyhedronData[{"Prism",6},"Dual"]

PolyhedronData[{"Prism",6},"Midsphere"]

PolyhedronData[{"Prism",6}]

PolyhedronData[{"CanonicalDipyramid",6}]