A triangle is the simplest possible closed polygon — three points connected by three straight sides. Despite this simplicity, triangles are the foundation of trigonometry, the basis of structural engineering's strongest and most stable shape, and the subject of an extraordinarily rich collection of geometric theorems studied continuously since antiquity.
A triangle is simply three straight lines connecting three points. It's the simplest possible shape that encloses any area at all — you need at least three sides to close off a region of space. But this simplicity hides remarkable structural strength: unlike a square or rectangle, which can be pushed sideways into a slanted parallelogram without changing the length of any side, a triangle's shape is completely rigid — the three side lengths alone completely and uniquely determine the triangle's exact shape.
In ordinary flat (Euclidean) geometry, the three interior angles of any triangle always add up to exactly 180° — no matter how the triangle's specific shape is stretched or distorted. This simple, elegant fact is one of the very first serious theorems students typically encounter in geometry class, and it turns out to be a defining, characteristic feature of flat space itself (see the Euclidean Geometry and Non-Euclidean Geometry entries — this angle sum rule actually fails in curved geometries).
Two triangles are congruent (identical in shape and size) if they satisfy any one of several standard criteria: SSS (all three corresponding sides equal), SAS (two sides and the angle between them equal), ASA (two angles and the side between them equal), or AAS (two angles and a non-included side equal). Notably, "AAA" (all three angles equal) is not sufficient for congruence — it only guarantees the two triangles are similar (same shape, but potentially different overall size).
Every triangle has several notable special points, each constructed differently and each with distinct geometric significance:
Remarkably, the centroid, circumcentre, and orthocentre always lie on a single straight line, called the Euler line (discovered by Euler, 1765) — a beautiful and initially quite unexpected geometric fact connecting three seemingly quite differently-constructed special points.
For a triangle with sides a, b, c opposite angles A, B, C respectively, the Law of Sines states a/sin(A) = b/sin(B) = c/sin(C), and the Law of Cosines generalises the Pythagorean theorem to non-right triangles: c² = a² + b² − 2ab·cos(C). These two laws together allow any triangle to be fully and completely solved (finding all sides and angles) from just a limited, minimal set of known initial measurements.
Beyond the basic ½×base×height formula, Heron's formula computes a triangle's area directly from its three side lengths alone: Area = √[s(s−a)(s−b)(s−c)], where s is the semi-perimeter (a+b+c)/2. This is the special case d=0 of Brahmagupta's more general formula for cyclic quadrilaterals (see the Brāhmasphuṭasiddhānta entry).
A particularly elegant classical result states that nine specific and quite differently-constructed points associated with any given triangle — the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments connecting each vertex to the orthocentre — all lie on a single common circle, called the nine-point circle (proved independently by several 19th-century geometers, including Karl Wilhelm Feuerbach, 1822). This circle's centre lies exactly at the midpoint of the segment connecting the circumcentre and orthocentre, and its radius is always exactly half that of the triangle's circumscribed circle.
The practical technique of triangulation — determining unknown distances and positions by measuring the angles within a network of triangles connecting known reference points — has been used in land surveying since antiquity, and remains fundamental to modern GPS positioning systems, which use triangulation-like calculations involving distances measured from multiple orbiting satellites. In computer graphics, triangle meshes — surfaces built entirely out of many small connected flat triangles — are the standard, near-universal way that 3D computer models and video game environments are actually represented internally and rendered onscreen, precisely because triangles are always guaranteed to be flat (any three points always determine a single, unique flat plane) and computationally very simple and efficient to render at scale.
As introduced in the Non-Euclidean Geometry discussion (see the Euclidean Geometry entry), triangles drawn on curved surfaces behave quite differently from ordinary flat ones. On a sphere, a triangle's three angles always sum to more than 180°, with the specific excess amount being directly and precisely proportional to the triangle's enclosed area (a foundational and elegant result in spherical trigonometry, essential to astronomical calculations, celestial navigation, and geodesy). On a hyperbolic surface, triangle angles instead always sum to less than 180°. This systematic angle-sum deviation from exactly 180° provides one of the simplest, most direct, and most intuitively accessible ways to experimentally and empirically detect and precisely measure a surface's intrinsic curvature.
In the study of symmetry (see the Group entry), a triangle group is generated by reflections in the three sides of a triangle with specific particular angles. These specific groups classify all the possible ways triangles of a given fixed shape can be used to perfectly and completely tile (tessellate) either the ordinary flat plane, a sphere, or the hyperbolic plane — connecting elementary triangle geometry directly and surprisingly deeply to abstract group theory, and playing a significant role in M.C. Escher's famous and mathematically sophisticated hyperbolic tessellation artworks.