Pyramidal Frustum

Author

Eric W. Weisstein
January 27, 2011

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

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Graphics

c={{{.6,0},{-.6,0},{0,-.7}},{{-1,-1},{1,-1},{1,1},{-1,1}},
Table[{Cos[2Pi i/5],Sin[2Pi i/5]},{i,5}],
	{{2,1.7},{-1.2,.9},{-.2,-.5},{.6,-.5}}};
s={.3,.3,.4,.5};

Show[GraphicsArray[
Graphics3D[#,Boxed->False]&/@Polyhedron/@Frustum@@@Transpose[{c,Table[1,{4}],s}]
]]

Show[GraphicsArray[
Graphics3D[#,Boxed->False]&/@Wireframe/@Polyhedron/@Frustum@@@Transpose[{c,Table[1,{4}],s}]
]]

V6

c={{{.6,0},{-.6,0},{0,-.7}},{{-1,-1},{1,-1},{1,1},{-1,1}},
Table[{Cos[2Pi i/5],Sin[2Pi i/5]},{i,5}],
	{{2,1.7},{-1.2,.9},{-.2,-.5},{.6,-.5}}};
s={.3,.3,.4,.5};

GraphicsRow[Graphics3D[#,Boxed->False]&/@Polyhedron/@Frustum@@@Transpose[{c,Table[1,{4}],s}]]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

Do[
	WriteLiveForm["pyrfrus"<>ToString[i]<>".m",
	Show[NonIntersectingPolygons[Frustum[c[[i]],1,s[[i]]]],Boxed->False,DisplayFunction->Identity]],
	{i,4}]

Table[Show[Graphics3D[Frustum[c[[i]],1,s[[i]]]],Boxed->False],{i,4}]

Surface Area and Volume

s[n_,a_,b_]:=Sqrt[Csc[Pi/n]^2/4(a-b)^2+h^2]
S[n_,a_,b_]:=1/2 n(a+b)s[n,a,b]
A[n_,a_]:=1/4 n a^2Cot[Pi/n]
V[n_,a_,b_]:=1/3 h(A[n,a]+A[n,b]+Sqrt[A[n,a]A[n,b]])

FullSimplify[PowerExpand[Table[{S[n,a,b],V[n,a,b]},{n,3,4}]]]

PowerExpand[V[4,a,a]]