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
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.
When a triangle has two equal sides, it is isosceles. It must also have two equal angles.
A triangle with three unique side lengths is known as a scalene triangle.
A triangle with all three angles smaller than 90 degrees is called an acute triangle. All equilateral triangles are also acute triangles.
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.
A triangle where at least one angle is greater than 90 degrees is called an obtuse triangle.
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.
The triangle has three Dynkin type symbols:
- x3o (regular)
- ox&#x (isosceles)
- ooo&#x (scalene)
Structure and Sections
As seen from a vertex, a triangle starts as a point that expands into a line segment.
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 =
- Vertex radius:
- Edge radius:
- Vertex angle: 60º
The simplest way to obtain vertex coordinates for an equilateral triangle is as a face of a 3D octahedron, thus yielding the coordinates as
thus giving a triangle of size
To obtain a triangle in the 2D plane with side 2, the coordinates are
- Tapertopic notation:
- Dual: self dual
- Vertex figure: Line segment, length 1
|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)|