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 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 =
- Surface area =
Subfacets
- 1 null polytope (-1D)
- 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
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Zerogon | 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 (-gon) | Pseudogon (-gon) |
Regular |
Rectified |
Truncated |
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Triangle | Triangle | Hexagon |