FANDOM



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 any 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 First Zeroth First Second Third Fourth Fifth Sixth Seventh Eighth Ninth Tenth Eleventh Twelfth Thirteenth Fourteenth Fifteenth Sixteenth ... Omegath
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

\{3^{\aleph_0}\}

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

\{3^{\aleph_0}, 4\}

Hydrotopes

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

Pentagon

\{5\}

Icosahedron

\{3, 5\}

Hexacosichoron

\{3^2, 5\}

Order-5 pentachoric honeycomb

\{3^3, 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

\{4, 3^{\aleph_0}\}

Cosmotopes

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

Pentagon

\{5\}

Dodecahedron

\{5, 3\}

Hecatonicosachoron

\{5, 3^2\}

Order-3 hecatonicosachoric honeycomb

\{5, 3^3\}

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\} ... \{\aleph_0\}
\{\frac{n}{1}\} Monogon Digon Triangle Square Pentagon Hexagon Heptagon Octagon Enneagon Decagon Hendecagon Dodecagon Tridecagon Tetradecagon Pentadecagon Hexadecagon ... Apeirogon
\{\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
\{\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
\{\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
\{\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
\{\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
\{\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
Regular
t_0 \{3\}
Rectified
t_1 \{3\}
Truncated
t_{0,1} \{3\}
Triangle Triangle Hexagon