3gon

A 3-gon, or triangle, is a polygon with three sides and three vertices. The term 3-gon is a formal mathematical designation, while triangle is the common name used in geometry and everyday language. When a three-sided figure lies in a plane or is interpreted in space, it is still a triangle as long as its three sides connect to form a closed figure.

In Euclidean geometry, the interior angles of a non-degenerate triangle sum to 180 degrees. If the three points are collinear, the figure is degenerate and has zero area. The area can be found as base times height divided by two, or by Heron’s formula from the side lengths a, b, c: area = sqrt[s(s-a)(s-b)(s-c)], where s is the semiperimeter (a+b+c)/2.

Types of triangles include classifications by sides—equilateral (three equal sides), isosceles (two equal sides), scalene (all sides different)—and by angles—acute (all angles less than 90°), right (one angle exactly 90°), obtuse (one angle greater than 90°). Common special triangles include the 30-60-90 and the 45-45-90 right triangles.

Triangles are central to trigonometry, linking side lengths to angles through sine, cosine, and tangent. The Pythagorean theorem applies to right triangles: a^2 + b^2 = c^2. In practice, triangles are fundamental in geometry, construction, and computer graphics, where complex shapes are built from triangular facets in meshes and tessellations.

In non-Euclidean contexts, the angle sum of a triangle differs: spherical triangles have sums greater than 180 degrees, while hyperbolic triangles have sums less than 180 degrees.