triangles

A triangle is a polygon with three edges and three vertices. The sum of its interior angles is 180 degrees. The triangle inequality states that the sum of the lengths of any two sides is greater than the third, which must hold for any nondegenerate triangle.

By side, triangles are classified as equilateral (three equal sides), isosceles (two equal sides), or scalene (no equal sides). By angle, they are classified as acute (all angles less than 90 degrees), right (one angle exactly 90 degrees), or obtuse (one angle greater than 90 degrees).

The perimeter of a triangle is the sum of its side lengths. The area can be found as base times height divided by two; equivalently, area = 1/2 ab sin C for sides a and b with included angle C. Heron's formula gives area from all three sides: with s = (a + b + c)/2, area = sqrt(s(s − a)(s − b)(s − c)).

Triangles are studied for congruence and similarity. Two triangles are congruent if corresponding sides and angles match, using criteria such as SSS, SAS, ASA, AAS, or RHS for right triangles. Similar triangles have equal corresponding angles and proportional corresponding sides.

Key centers associated with a triangle include the centroid (intersection of medians, located 2:1 along each median from vertex to midpoint of the opposite side), the circumcenter (intersection of perpendicular bisectors), the incenter (intersection of angle bisectors), and the orthocenter (intersection of altitudes).

In geometry and applications, triangles underpin distance measurements, angle calculations, and area computations, and they appear in surveying, architecture, computer graphics, and trigonometry.