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 any 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
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