Triangle (EntityTopic, 17)
From Hi.gher. Space
A triangle is a two-dimensional simplex.
Equations
- The area of an equilateral triangle with side length l is:
^{√3}∕_{4} · l^{2}
- The area of an arbitrary triangle, where a and b are two sides of the triangle and C is the angle between them, is:
^{1}∕_{2} · ab · sin C
Dissection
The equilateral triangle of side 1 may be dissected into 3× triangle with sides 1, 2×3^{-2-1} and angles 2×30°, 120°.
Incidence matrix
Usage as facets
- 20× 1-facets of a icosahedron
- 8× 1-facets of a octahedron
- pyramid: 4× 1-facets of a tetrahedron
- 20× 1-facets of a rhodomesohedron
- 8× 1-facets of a stauromesohedron
- 20× 1-facets of a decagonal bipyramid
- 12× 1-facets of a hexagonal bipyramid
- 16× 1-facets of a octagonal bipyramid
- 10× 1-facets of a pentagonal bipyramid
- 60× 1-facets of a pentakis dodecahedron
- 8× 1-facets of a square antiprism
- 24× 1-facets of a tetrakis hexahedron
- 60× 1-facets of a triakis icosahedron
- 24× 1-facets of a triakis octahedron
- 12× 1-facets of a triakis tetrahedron
- 6× 1-facets of a triangular bipyramid
- prism: 2× 1-facets of a triangular prism
- 8× 1-facets of a truncated cube
- 20× 1-facets of a truncated dodecahedron
- 4× 1-facets of a truncated tetrahedron
- 5× 1-facets of a pentagonal pyramid
- 20× 1-facets of a rhodoperihedron
- 4× 1-facets of a square pyramid (named sweep: edges)
- 8× 1-facets of a stauroperihedron
- 48× 1-facets of a disdyakis dodecahedron
- 120× 1-facets of a disdyakis triacontahedron
- 24× 1-facets of a snub cube
- 8× 1-facets of a snub cube
- 60× 1-facets of a snub dodecahedron
- 20× 1-facets of a snub dodecahedron
- 4× 1-facets of a parabiorthotriangulated cube
- 4× 1-facets of a parabiorthotriangulated cube_
- 8× 1-facets of a snub disphenoid
- 4× 1-facets of a snub disphenoid
- 2× 1-facets of a triaugmented triangular prism (named end)
- 6× 1-facets of a triaugmented triangular prism (named mid)
- 6× 1-facets of a triaugmented triangular prism (named side)
- 6× 1-facets of a J3 dual
- 8× 1-facets of a J4 dual
- 10× 1-facets of a J5 dual
- 5× 1-facets of a pentagonal cupola
- 2× 1-facets of a rhombocupolawedge
- 4× 1-facets of a rhombocupolawedge
- 2× 1-facets of a rhombopyrawedge
- 4× 1-facets of a square cupola
- 1× 1-facets of a triangular cupola
- 3× 1-facets of a triangular cupola
- 8× 1-facets of a J37 (named triangles)
- 3× 1-facets of a J63 dual
- 3× 1-facets of a J63 dual
- 1× 1-facets of a tridiminished icosahedron (named top)
- 3× 1-facets of a tridiminished icosahedron (named lat)
- 1× 1-facets of a tridiminished icosahedron (named bot)
- 4× 1-facets of a bilunabirotunda (named cross)
- 4× 1-facets of a bilunabirotunda (named end)
- 4× 1-facets of a J91 dual
- 2× 1-facets of a triangulated triangular prism
- 2× 1-facets of a triangulated triangular prism
- 2× 1-facets of a hexagonal cupolawedge
- 2× 1-facets of a hexagonal cupolawedge
- 4× 1-facets of a hexagonal pyrawedge
- 2× 1-facets of a hexagonal pyrawedge
- 2× 1-facets of a hexagonal pyrawedge
- 2× 1-facets of a (dual of metabidiminished icosahedron)
- 1× 1-facets of a digon-unpinched square pyramid
- 2× 1-facets of a digon-unpinched square pyramid
- 2× 1-facets of a digon-unpinched square pyramid
- 2× 1-facets of a metabidiminished icosahedron (named roo)
- 2× 1-facets of a metabidiminished icosahedron (named oog)
- 4× 1-facets of a metabidiminished icosahedron (named ogb)
- 2× 1-facets of a metabidiminished icosahedron (named ggb)
- 2× 1-facets of a metabigyrotriangulated cube
- 2× 1-facets of a metabigyrotriangulated cube
- 2× 1-facets of a transposed diminished cube
- 2× 1-facets of a transposed diminished cube
- 1× 1-facets of a triangular hebesphenorotunda (named top)
- 3× 1-facets of a triangular hebesphenorotunda (named tola)
- 6× 1-facets of a triangular hebesphenorotunda (named bola)
- 3× 1-facets of a triangular hebesphenorotunda (named lat)
- 2× 1-facets of a biunpinched cube
- 1× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 2× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 2× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 2× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 1× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 1× 1-facets of a parabiorthotriangulated orthopinched triangular cupola
- 2× 1-facets of a pentagonal cupolawedge
- 2× 1-facets of a pentagonal cupolawedge
- 2× 1-facets of a pentagonal cupolawedge
- 1× 1-facets of a pentagonal cupolawedge
- 2× 1-facets of a pentagonal transcupolawedge
- 2× 1-facets of a pentagonal transcupolawedge
- 2× 1-facets of a pentagonal transcupolawedge
- 1× 1-facets of a pentagonal transcupolawedge
- 2× 1-facets of a square cupolawedge
- 2× 1-facets of a square transcupolawedge (named ypg)
- 2× 1-facets of a square transcupolawedge (named ocb)
- 1× 1-facets of a triangulated pinched triangular cupola (named lime)
- 1× 1-facets of a triangulated pinched triangular cupola (named green)
- 1× 1-facets of a triangulated pinched triangular cupola (named cyan)
- 1× 1-facets of a triangulated pinched triangular cupola (named blue)
- 1× 1-facets of a triangulated pinched triangular cupola (named pink)
- 32× 2-facets of a aerochoron
- 1200× 2-facets of a hydrochoron
- 10× 2-facets of a pyrochoron
- 96× 2-facets of a xylochoron
- 6× 2-facets of a duotrianglinder
- 18× 2-facets of a duotrianglone
- 30× 2-facets of a (dual of pyrorectichoron)
- 20× 2-facets of a pyrorectichoron (named tet-oct)
- 10× 2-facets of a pyrorectichoron (named oct-oct)
- 24× 2-facets of a (dual of octahedral prism)
- 12× 2-facets of a (dual of tetrahedral prism)
- 4× 2-facets of a (dual of tetrahedral prism)
- 48× 2-facets of a bixylodiminished hydrochoron (named verticals)
- 72× 2-facets of a bixylodiminished hydrochoron (named laterals)
- 16× 2-facets of a octahedral prism
- 8× 2-facets of a tetrahedral prism
- 144× 2-facets of a trixylodiminished hydrochoron
- 12× 2-facets of a (dual of triangular diprism)
- 12× 2-facets of a (dual of triangular diprism)
- 12× 2-facets of a cubic pyramid (named sweep: edges)
- 12× 2-facets of a octahedral pyramid
- 8× 2-facets of a octahedral pyramid
- 4× 2-facets of a triangular diprism (named joins 2 triangular prisms)
- 18× 2-facets of a (dual of triangular octagoltriate) (named isosceles triangle)
- 6× 2-facets of a triangular octagoltriate (named small triangles)
- 8× 2-facets of a (dual of square pyramid prism)
- 8× 2-facets of a (dual of square pyramid prism)
- 4× 2-facets of a (dual of square pyramid prism)
- 8× 2-facets of a square pyramid prism
- 8× 2-facets of a gyrated octahedral prism (named walls)
- 8× 2-facets of a gyrated octahedral prism (named wall corners)
- 16× 2-facets of a K4.13 dual
- 4× 2-facets of a K4.8
- 4× 2-facets of a K4.8
- 4× 2-facets of a K4.8
- 4× 2-facets of a K4.8 dual
- 8× 2-facets of a K4.8 dual
- 4× 2-facets of a K4.8 dual
- 4× 2-facets of a (dual of square biantiprismatic ring)
- 16× 2-facets of a (dual of square biantiprismatic ring)
- 8× 2-facets of a (dual of square biantiprismatic ring)
- 8× 2-facets of a square biantiprismatic ring (named lace end)
- 8× 2-facets of a square biantiprismatic ring (named lace side)
- 8× 2-facets of a square biantiprismatic ring (named lace mid)
- 288× 2-facets of a (dual of rectified snub demitesseract)
- 288× 2-facets of a (dual of rectified snub demitesseract)
- 576× 2-facets of a (dual of rectified snub demitesseract)
- 288× 2-facets of a (dual of rectified snub demitesseract)
- 96× 2-facets of a rectified snub demitesseract
- 96× 2-facets of a rectified snub demitesseract
- 288× 2-facets of a rectified snub demitesseract
- 288× 2-facets of a rectified snub demitesseract
- 96× 2-facets of a rectified snub demitesseract
- 96× 2-facets of a rectified snub demitesseract
- 40× 2-facets of a castellated rhodoperihedral prism
- 120× 2-facets of a castellated rhodoperihedral prism
- 120× 2-facets of a castellated rhodoperihedral prism
- 120× 2-facets of a D4.3.1 dual
- 120× 2-facets of a D4.3.1 dual
- 120× 2-facets of a D4.3.1 dual
- 120× 2-facets of a D4.3.1 dual
- 288× 2-facets of a (dual of truncated snub demitesseract)
- 288× 2-facets of a (dual of truncated snub demitesseract)
- 288× 2-facets of a (dual of truncated snub demitesseract)
- 576× 2-facets of a (dual of truncated snub demitesseract)
- 288× 2-facets of a (dual of truncated snub demitesseract)
- 96× 2-facets of a truncated snub demitesseract
- 288× 2-facets of a truncated snub demitesseract
- 96× 2-facets of a truncated snub demitesseract
- 32× 2-facets of a D4.11
- 96× 2-facets of a D4.11
- 32× 2-facets of a D4.11
- 32× 2-facets of a D4.11
- 96× 2-facets of a D4.11
- 96× 2-facets of a D4.11
- 32× 2-facets of a D4.11
- 32× 2-facets of a D4.11
- 96× 2-facets of a D4.11 dual
- 192× 2-facets of a D4.11 dual
- 96× 2-facets of a D4.11 dual
- 96× 2-facets of a D4.11 dual
- 4× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 2× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 8× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 8× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 4× 2-facets of a bilunabirotunda pseudopyramid (named cross)
- 4× 2-facets of a bilunabirotunda pseudopyramid (named diagonals)
- 4× 2-facets of a bilunabirotunda pseudopyramid (named horz-to-roof)
- 2× 2-facets of a bilunabirotunda pseudopyramid (named cross-to-roof)
- 8× 2-facets of a bilunabirotunda pseudopyramid (named diag-to-roof)
- 2× 2-facets of a bilunabirotunda pseudopyramid (named end-to-roof)
- 8× 2-facets of a bilunabirotunda pseudopyramid (named cross-cuboid-roof)
- 120× 2-facets of a castellated rhodopantohedral prism (named 3 diag)
- 20× 2-facets of a castellated rhodopantohedral prism (named 3 horz)
- 120× 2-facets of a castellated rhodopantohedral prism (named 5 diag)
- 6× 2-facets of a D4.16 (named digon to outermost)
- 2× 2-facets of a D4.16 (named equatorial caps)
- 6× 2-facets of a D4.16 (named equatorial cap-petals)
- 12× 2-facets of a D4.16 (named equatorial sweep-petals)
- 6× 2-facets of a D4.16 (named equatorial sweeps)
- 6× 2-facets of a D4.16 (named cap guard)
- 6× 2-facets of a D4.16 (named sweep guard)
- 2× 2-facets of a D4.16 (named ends of trip)
- 6× 2-facets of a D4.16 dual
- 12× 2-facets of a D4.16 dual
- 12× 2-facets of a D4.16 dual
- 6× 2-facets of a D4.16 dual
- 6× 2-facets of a D4.16 dual
- 1× 2-facets of a D4.16 dual
- 120× 2-facets of a D4.3.2 dual
- 60× 2-facets of a D4.3.2 dual
- 240× 2-facets of a D4.3.2 dual
- 240× 2-facets of a D4.3.2 dual
- 120× 2-facets of a D4.3.2 dual
- 120× 2-facets of a D4.3.2 dual
- 8× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 8× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 8× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 8× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 2-facets of a (dual of bitrigonal diminished pyrocantichoron)
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 8× 2-facets of a bitrigonal diminished pyrocantichoron
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 16× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 32× 2-facets of a (4D analog of J37)
- 16× 2-facets of a (4D analog of J37)
- 8× 2-facets of a (dual of 4D analog of J37)
- 8× 2-facets of a (dual of 4D analog of J37)
- 16× 2-facets of a (dual of 4D analog of J37)
- 32× 2-facets of a (dual of 4D analog of J37)
- 32× 2-facets of a (dual of 4D analog of J37)
- 32× 2-facets of a (dual of 4D analog of J37)
- 32× 2-facets of a (dual of 4D analog of J37)
- 32× 2-facets of a (dual of 4D analog of J37)
- 32× 2-facets of a (dual of 4D analog of J37)
- 16× 2-facets of a (dual of 4D analog of J37)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 6× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 3× 2-facets of a (dual of triangular hebesphenorotunda pseudopyramid)
- 1× 2-facets of a triangular hebesphenorotunda pseudopyramid (named top)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named tola)
- 6× 2-facets of a triangular hebesphenorotunda pseudopyramid (named bola)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named lat)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named top-pen to apex)
- 6× 2-facets of a triangular hebesphenorotunda pseudopyramid (named pen-tola to apex)
- 6× 2-facets of a triangular hebesphenorotunda pseudopyramid (named pen-bola to apex)
- 6× 2-facets of a triangular hebesphenorotunda pseudopyramid (named bola-square to apex)
- 6× 2-facets of a triangular hebesphenorotunda pseudopyramid (named bola-lat to apex)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named lat-bot to apex)
- 1× 2-facets of a triangular hebesphenorotunda pseudopyramid (named roof)
- 3× 2-facets of a triangular hebesphenorotunda pseudopyramid (named top to roof)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 6× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named bpp)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named ggb)
- 24× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named obp orb)
- 24× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named obp rad)
- 24× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named ogb orb)
- 24× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named ogb rad)
- 24× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named ogg)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named oog orb)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named oog rad)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named oob)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named roo vert)
- 12× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named roo horz)
- 2× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named rrr)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 24× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 6× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 2× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named rrr - top)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named roo horz - tola)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named roo vert)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oob)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oog rad)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named oog orb)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named ogg)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named ogb rad)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named ogb orb)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named obp rad)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named obp orb - bola)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named ggb)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named bpp - lat)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named top-pen to apex)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named pen-tola to apex)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named pen-bola to apex)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named bola-square to apex)
- 24× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named bola-lat to apex)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named lat-bot to apex)
- 4× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named roof)
- 12× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named top to roof)
- 12× 2-facets of a D4.7 (named roo)
- 24× 2-facets of a D4.7 (named roy)
- 24× 2-facets of a D4.7 (named rob)
- 24× 2-facets of a D4.7 (named ryg)
- 24× 2-facets of a D4.7 (named ryc)
- 24× 2-facets of a D4.7 (named rgc)
- 12× 2-facets of a D4.7 (named rcc)
- 12× 2-facets of a D4.7 (named oob)
- 24× 2-facets of a D4.7 (named ygc)
- 12× 2-facets of a D4.7 (named gcc)
- 12× 2-facets of a D4.7 (named ccb)
- 12× 2-facets of a D4.7 (named ccs parl)
- 12× 2-facets of a D4.7 (named ccs perp)
- 24× 2-facets of a D4.7 (named cbp)
- 24× 2-facets of a D4.7 (named cps)
- 12× 2-facets of a D4.7 (named bpp)
- 8× 2-facets of a D4.7 (named ppp)
- 12× 2-facets of a D4.7 (named pps)
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 12× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 24× 2-facets of a D4.7 dual
- 80× 3-facets of a aeroteron
- 20× 3-facets of a pyroteron
- 36× 3-facets of a (dual of triangular triprism)
- 18× 3-facets of a (dual of triangular triprism)
- 8× 3-facets of a (dual of triangular triprism)
- 8× 3-facets of a triangular triprism (named joins 3 triangular prisms)
Simplices |
triangle • tetrahedron • pyrochoron • pyroteron • pyropeton |
Notable Dishapes | |
Flat: | triangle • square • pentagon • hexagon • octagon • decagon |
Curved: | circle |
3. 11 Square | 4. 1^{1} Triangle | 5. 3 Sphere |
List of tapertopes |