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A triangle is the 2 simplex. The study of triangles is called trigonometry (plus the study of angles). Its Bowers acronym is "trig". It is also known as a pyrogon under the elemental naming scheme.

Types of Triangle

By Sides

Equilateral

When all of the angles and edges of a triangle are equal a triangle is equilateral. Each angle must have a measure of exactly 60 degrees.

Isosceles

When a triangle has two equal sides, it is isosceles. It must also have two equal angles.

Scalene

A triangle with three unique side lengths is known as a scalene triangle.

By Angles

Acute

A triangle with all three angles smaller than 90 degrees is called an acute triangle. All equilateral triangles are also acute triangles.

Right-Angled

A triangle with one right angle is a right-angled triangle. Their side lengths follow the equation $ a^2 + b^2 = c^2 $, where c is the edge length of the side opposite the right angle.

Obtuse

A triangle where at least one angle is greater than 90 degrees is called an obtuse triangle.

Special Cases

A triangle with more than one right angle is usually degenerate but can appear on the surface of a sphere. The same applies to a triangle with angles that sum to greater than 180 degrees, and to triangles that have an angle equal to 180 degrees.

Symbols

The triangle has three Dynkin type symbols:

  • x3o (regular)
  • ox&#x (isosceles)
  • ooo&#x (scalene)

Structure and Sections

Sections

As seen from a vertex, a triangle starts as a point that expands into a line segment.

Hypervolumes

Formulas for a general triangle

If we let a, b, c be the sides of the triangle, and h be the higher perpendicular to side a, then we can obtain these formulae:

  • Edge length = $ a+b+c $
  • Surface area = $ \frac{1}{2}ah $

Subfacets

Radii

  • Vertex radius: $ \frac{\sqrt{3}{3}}l $
  • Edge radius: $ \frac{\sqrt{3}{6}}l $

Angles

  • Vertex angle: 60º

Vertex coordinates

The simplest way to obtain vertex coordinates for an equilateral triangle is as a face of a 3D octahedron, thus yielding the coordinates as

  • (1,0,0)
  • (0,1,0)
  • (0,0,1)

thus giving a triangle of size $ \sqrt{2} $.

To obtain a triangle in the 2D plane with side 2, the coordinates are

  • (±1,-√3/3)
  • (0,2√3/3)

Notations

  • Tapertopic notation: $ 1^1 $

Related shapes

See Also

Dimensionality Negative One Zero One Two Three Four Five Six Seven Eight Nine Ten Eleven Twelve Thirteen Fourteen Fifteen Sixteen ... Aleph null
Simplex

$ \{3^{n-1}\} $

Null polytope

$ \emptyset $

Point

$ () $
$ \mathbb{B}^0 $

Line segment

$ \{\} $
$ \mathbb{B}^1 $

Triangle

$ \{3\} $

Tetrahedron

$ \{3^2\} $

Pentachoron

$ \{3^3\} $

Hexateron

$ \{3^4\} $

Heptapeton

$ \{3^5\} $

Octaexon

$ \{3^6\} $

Enneazetton

$ \{3^7\} $

Decayotton

$ \{3^8\} $

Hendecaxennon

$ \{3^9\} $

Dodecadakon

$ \{3^{10}\} $

Tridecahendon

$ \{3^{11}\} $

Tetradecadokon

$ \{3^{12}\} $

Pentadecatradakon

$ \{3^{13}\} $

Hexadecatedakon

$ \{3^{14}\} $

Heptadecapedakon

$ \{3^{15}\} $

... Omegasimplex
Cross

$ \{3^{n-2},4\} $

Square

$ \{4\} $

Octahedron

$ \{3, 4\} $

Hexadecachoron

$ \{3^2, 4\} $

Pentacross

$ \{3^3, 4\} $

Hexacross

$ \{3^4, 4\} $

Heptacross

$ \{3^5, 4\} $

Octacross

$ \{3^6, 4\} $

Enneacross

$ \{3^7, 4\} $

Dekacross

$ \{3^8, 4\} $

Hendekacross

$ \{3^9, 4\} $

Dodekacross

$ \{3^{10}, 4\} $

Tridekacross

$ \{3^{11}, 4\} $

Tetradekacross

$ \{3^{12}, 4\} $

Pentadekacross

$ \{3^{13}, 4\} $

Hexadekacross

$ \{3^{14}, 4\} $

... Omegacross
Hydrotopes

$ \{3^{n-2}, 5\} $

Pentagon

$ \{5\} $

Icosahedron

$ \{3, 5\} $

Hexacosichoron

$ \{3^2, 5\} $

Order-5 pentachoric tetracomb

$ \{3^3, 5\} $

Order-5 hexateric pentacomb

$ \{3^4, 5\} $

...
Hypercube

$ \{4, 3^{n-2}\} $

Square

$ \{4\} $

Cube

$ \{4, 3\} $

Tesseract

$ \{4, 3^2\} $

Penteract

$ \{4, 3^3\} $

Hexeract

$ \{4, 3^4\} $

Hepteract

$ \{4, 3^5\} $

Octeract

$ \{4, 3^6\} $

Enneract

$ \{4, 3^7\} $

Dekeract

$ \{4, 3^8\} $

Hendekeract

$ \{4, 3^9\} $

Dodekeract

$ \{4, 3^{10}\} $

Tridekeract

$ \{4, 3^{11}\} $

Tetradekeract

$ \{4, 3^{12}\} $

Pentadekeract

$ \{4, 3^{13}\} $

Hexadekeract

$ \{4, 3^{14}\} $

... Omegeract
Cosmotopes

$ \{5, 3^{n-2}\} $

Pentagon

$ \{5\} $

Dodecahedron

$ \{5, 3\} $

Hecatonicosachoron

$ \{5, 3^2\} $

Order-3 hecatonicosachoric tetracomb

$ \{5, 3^3\} $

Order-3-3 hecatonicosachoric pentacomb

$ \{5, 3^4\} $

...
Hyperball

$ \mathbb B^n $

Disk

$ \mathbb B^2 $

Ball

$ \mathbb B^3 $

Gongol

$ \mathbb B^4 $

Pentorb

$ \mathbb B^5 $

Hexorb

$ \mathbb B^6 $

Heptorb

$ \mathbb B^7 $

Octorb

$ \mathbb B^8 $

Enneorb

$ \mathbb B^9 $

Dekorb

$ \mathbb B^{10} $

Hendekorb

$ \mathbb B^{11} $

Dodekorb

$ \mathbb B^{12} $

Tridekorb

$ \mathbb B^{13} $

Tetradekorb

$ \mathbb B^{14} $

Pentadekorb

$ \mathbb B^{15} $

Hexadekorb

$ \mathbb B^{16} $

... Omegaball

$ \mathbb B^{\aleph_0} $

Regular polygons $ \{1\} $ $ \{2\} $ $ \{3\} $ $ \{4\} $ $ \{5\} $ $ \{6\} $ $ \{7\} $ $ \{8\} $ $ \{9\} $ $ \{10\} $ $ \{11\} $ $ \{12\} $ $ \{13\} $ $ \{14\} $ $ \{15\} $ $ \{16\} $ ... $ \{\infty\} $ $ \{\frac{\pi i}{\lambda}\} $
$ \{\frac{n}{1}\} $ Monogon Digon Triangle Square Pentagon Hexagon Heptagon Octagon Enneagon Decagon Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon ... Apeirogon Pseudogon ($ \frac{\pi i}{\lambda} $-gon)
$ \{\frac{n}{2}\} $ N/A N/A Triangle (retrograde) Degenerate Pentagram Degenerate Heptagram Degenerate Enneagram Degenerate Small hendecagram Degenerate Small tridecagram Degenerate Small pentadecagram Degenerate ... N/A Pseudogon ($ \frac{\pi i}{2\lambda} $-gon)
$ \{\frac{n}{3}\} $ N/A N/A N/A Square (retrograde) Pentagram (retrograde) Degenerate Great heptagram Octagram Degenerate Decagram Hendecagram Degenerate Tridecagram Tetradecagram Degenerate Small hexadecagram ... N/A Pseudogon ($ \frac{\pi i}{3\lambda} $-gon)
$ \{\frac{n}{4}\} $ N/A N/A N/A N/A Pentagon (retrograde) Degenerate Great heptagram (retrograde) Degenerate Great enneagram Degenerate Great hendecagram Degenerate Medial tridecagram Degenerate Pentadecagram Degenerate ... N/A Pseudogon ($ \frac{\pi i}{4\lambda} $-gon)
$ \{\frac{n}{5}\} $ N/A N/A N/A N/A N/A Hexagon (retrograde) Heptagram (retrograde) Octagram (retrograde) Great enneagram (retrograde) Degenerate Grand hendecagram Dodecagram Great tridecagram Great tetradecagram Degenerate Hexadecagram ... N/A Pseudogon ($ \frac{\pi i}{5\lambda} $-gon)
$ \{\frac{n}{6}\} $ N/A N/A N/A N/A N/A N/A Heptagon (retrograde) Degenerate Degenerate Degenerate Grand hendecagram (retrograde) Degenerate Grand tridecagram Degenerate Degenerate Degenerate ... N/A Pseudogon ($ \frac{\pi i}{6\lambda} $-gon)
$ \{\frac{n}{7}\} $ N/A N/A N/A N/A N/A N/A N/A Octagon (retrograde) Enneagram (retrograde) Decagram (retrograde) Great hendecagram (retrograde) Dodecagram (retrograde) Grand tridecagram (retrograde) Degenerate Great pentadecagram Great hexadecagram ... N/A Pseudogon ($ \frac{\pi i}{7\lambda} $-gon)
Regular
$ t_0 \{3\} $
Rectified
$ t_1 \{3\} $
Truncated
$ t_{0,1} \{3\} $
Triangle Triangle Hexagon