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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

• 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

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

$\{3^9\}$

$\{3^{10}\}$

Tridecahendon

$\{3^{11}\}$

$\{3^{12}\}$

$\{3^{13}\}$

$\{3^{14}\}$

$\{3^{15}\}$

... Omegasimplex
Cross

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

Square

$\{4\}$

Octahedron

$\{3, 4\}$

$\{3^2, 4\}$

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

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

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

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

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

... 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

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

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

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

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

... 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

$\mathbb B^{13}$

$\mathbb B^{14}$

$\mathbb B^{15}$

$\mathbb B^{16}$

... Omegaball

$\mathbb B^{\aleph_0}$

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