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An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
The relations can be made apparent by examining the vertex figures obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example, the cube has vertex figure 4.4.4, which is to say, three adjacent square faces. The possible faces are 3 - equilateral ...
The only plane triangles that tile the plane once over are (3 3 3), (4 2 4), and (3 2 6): they are respectively the equilateral triangle, the 45-45-90 right isosceles triangle, and the 30-60-90 right triangle. It follows that any plane triangle tiling the plane multiple times must be built up from multiple copies of one of these.
A regular hexagram, , can be seen as a compound composed of an upwards (blue here) and downwards (pink) facing equilateral triangle, with their intersection as a regular hexagon (in green). A hexagram or sexagram is a six-pointed geometric star figure with the Schläfli symbol {6/2}, 2{3}, or {{3}}.
Given a square, construct equilateral triangles on two adjacent edges, either both inside or both outside the square. Then the triangle formed by joining the vertex of the square distant from both triangles and the vertices of the triangles distant from the square is equilateral. [2]
Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal. [5] The first is made of 2-edges, and next two are triangular edges, and the last has overlapping hexagonal edges.
These tetrahedra cover their triangular base, and the resulting polyhedron has six triangles, five vertices, and nine edges. [3] A triangular bipyramid is said to be right if the tetrahedra are symmetrically regular and both of their apices are on a line passing through the center of the base; otherwise, it is oblique. [4] [5]