Triangle their internal angles, measured here in degrees. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a "2-simplex" (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4
Triangular number Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The th triangular number is the number of dots in the triangular arrangement with dots on a side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers , starting at the 0th triangular number, is The triangle numbers are given by the following explicit formulas: where formula_2 is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read