Triangle

The triangle, or trig, is the simplest possible polygon, excluding the degenerate digon. Its highest symmetry version is called an equilateral triangle, to emphasize its three equal side lengths. It's the two dimensional simplex.

The equilateral triangle is one of the only three regular polygons that can tile the plane, the other two being the square and the hexagon. Its tiling is called the triangular tiling. It's also the regular simplex of highest dimension that can tile its respective hyperplane.

This is one of two polygons without a stellation, the other being the square, and one of three without a non-compound stellation, the third being the hexagon.

Vertex coordinates
The vertices of a triangle of edge length 1 centered at the origin are:
 * (±1/2, –$\sqrt{3}$/6),
 * (0, $\sqrt{3}$/3).

In vertex figures
The equilateral triangle is seen in the vertex figures of three Platonic Solids. To see where other kinds of triangles appear as vertex figures, see their respective pages.

Other kinds of triangles
Beside the equilateral triangle, there are other kinds of triangles with non-equal edge lengths. These are the isosceles triangle with only two equal edge lengths, and the scalene triangle, with no equal edge lengths. Notably, these retain many of the properties of the highest-symmetry variant: any triangle is convex, has an inscribed and an exscribed circle, and tiles the plane. The first two properties don't generalize to any other polygon, and the third generalizes only to the quadrilateral.