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 , 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 =

- Surface area =

### Subfacets

- 3 points (0D)
- 3 line segments (1D)
- 1
**triangle**(2D)

### Radii

- Vertex radius:

- Edge radius:

### 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 .

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

- (±1,-√3/3)
- (0,2√3/3)

### Notations

- Tapertopic notation:

### Related shapes

- Dual: self dual
- Vertex figure: Line segment, length 1

## See Also

Regular polygons | ... | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Monogon | Digon | Triangle
| Square | Pentagon | Hexagon | Heptagon | Octagon | Enneagon | Decagon | Hendecagon | Dodecagon | Tridecagon | Tetradecagon | Pentadecagon | Hexadecagon | ... | Apeirogon | |

N/A | N/A | Triangle (retrograde)
| Degenerate | Pentagram | Degenerate | Heptagram | Degenerate | Enneagram | Degenerate | Small hendecagram | Degenerate | Small tridecagram | Degenerate | Small pentadecagram | Degenerate | ... | N/A | |

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 | |

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 | |

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 | |

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 | |

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 | Rectified | Truncated |
---|---|---|

Triangle
| Triangle
| Hexagon |