Elongated triangular pyramid Contents Formulae Related polyhedra and honeycombs References External links Navigation menu10.4153/cjm-1966-021-801855070132.14603Elongated triangular pyramidJohnson solidElongated triangular pyramideexpanding ite

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square pyramidpentagonal pyramidtriangular cupolasquare cupolapentagonal cupolapentagonal rotundaelongated triangular cupolaelongated square cupolaelongated pentagonal cupolaelongated pentagonal rotundagyroelongated triangular cupolagyroelongated square cupolagyroelongated pentagonal cupolagyroelongated pentagonal rotundagyrobifastigiumtriangular orthobicupolasquare orthobicupolasquare gyrobicupolapentagonal orthobicupolapentagonal gyrobicupolapentagonal orthocupolarotundapentagonal gyrocupolarotundapentagonal orthobirotundaelongated triangular orthobicupolaelongated triangular gyrobicupolaelongated square gyrobicupolaelongated pentagonal orthobicupolaelongated pentagonal gyrobicupolaelongated pentagonal orthocupolarotundaelongated pentagonal gyrocupolarotundaelongated pentagonal orthobirotundaelongated pentagonal gyrobirotundagyroelongated triangular bicupolagyroelongated square bicupolagyroelongated pentagonal bicupolagyroelongated pentagonal cupolarotundagyroelongated pentagonal birotundaaugmented dodecahedronparabiaugmented dodecahedronmetabiaugmented dodecahedrontriaugmented dodecahedronmetabidiminished icosahedrontridiminished icosahedronaugmented tridiminished icosahedronsnub disphenoidsnub square antiprismsphenocoronaaugmented sphenocoronasphenomegacoronahebesphenomegacoronadisphenocingulumbilunabirotundatriangular hebesphenorotunda


Johnson solidsSelf-dual polyhedraPyramids and bipyramidsPolyhedron stubs


geometryJohnson solidstetrahedrontriangular prismpyramiddualJohnson solidconvexpolyhedraregularuniformPlatonic solidsArchimedean solidsprismsantiprismsNorman Johnsonformulaevolumesurface areafacesregulartessellation of spacesquare pyramidsoctahedra






















Elongated triangular pyramid
Elongated triangular pyramid.png
Type
Johnson
J6 - J7 - J8
Faces1+3 triangles
3 squares
Edges12
Vertices7
Vertex configuration1(33)
3(3.42)
3(32.42)
Symmetry group
C3v, [3], (*33)
Rotation group
C3, [3]+, (33)
Dual polyhedronself
Propertiesconvex
Net
Elongated Triangular Pyramid Net.svg

In geometry, the elongated triangular pyramid is one of the Johnson solids (J7). As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base. Like any elongated pyramid, the resulting solid is topologically (but not geometrically) self-dual.


A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]




Contents





  • 1 Formulae

    • 1.1 Dual polyhedron



  • 2 Related polyhedra and honeycombs


  • 3 References


  • 4 External links




Formulae


The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[2]


V=(112(2+33))a3≈0.550864...a3displaystyle V=(frac 112(sqrt 2+3sqrt 3))a^3approx 0.550864...a^3


A=(3+3)a2≈4.73205...a2displaystyle A=(3+sqrt 3)a^2approx 4.73205...a^2


If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together.



Dual polyhedron


Topologically, the elongated triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces: one equilateral triangle, three isosceles triangles and three isosceles trapezoids.






Dual elongated triangular pyramid
Net of dual

Dual elongated triangular pyramid.png

Dual elongated triangular pyramid net.png


Related polyhedra and honeycombs


The elongated triangular pyramid can form a tessellation of space with square pyramids and/or octahedra.[3]



References




  1. ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.mw-parser-output cite.citationfont-style:inherit.mw-parser-output .citation qquotes:"""""""'""'".mw-parser-output .citation .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .citation .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-ws-icon abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center.mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-maintdisplay:none;color:#33aa33;margin-left:0.3em.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em.


  2. ^ Stephen Wolfram, "Elongated triangular pyramid" from Wolfram Alpha. Retrieved July 21, 2010.


  3. ^ http://woodenpolyhedra.web.fc2.com/J7.html




External links


  • Eric W. Weisstein, Johnson solid (Elongated triangular pyramid) at MathWorld.






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