Triangle (EntityTopic, 17)
From Hi.gher. Space
A triangle is a two-dimensional simplex.
Triangles are one of the three regular polygons that can tile the plain, forming the triangular tiling.
Coordinates
The Cartesian coordinates of an equilateral triangle centered at the origin with side length 2 are:
(-1, -√3/3), (1, -√3/3), (0, 2√3/3)
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, √3/3, √3/3 and angles 30°, 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 |