Equilateral triangle

Key properties

Several basic characteristics follow immediately from side and angle equality:

• All three sides are congruent and all three internal angles are congruent (60° each).
• It is a special case of an isosceles triangle (every equilateral triangle has three axes of symmetry).
• Altitudes, medians and perpendicular bisectors coincide for each vertex, meeting at a single central point (the centroid, which is also the circumcenter and incenter).
• Rotational symmetry of order three and three lines of reflection symmetry.

Metric formulas and constructions

Many common quantities for an equilateral triangle can be written in terms of the side length, usually denoted s. Useful formulas include:

• Height (altitude): h = (sqrt(3)/2)·s. This divides the triangle into two congruent 30°–60°–90° right triangles.
• Area: A = (sqrt(3)/4)·s². This follows from base·height/2 using the altitude above.
• Inradius (radius of inscribed circle): r = (s·sqrt(3))/6; Circumradius (radius of circumscribed circle): R = (s·sqrt(3))/3.

Equilateral triangles can be constructed with straightedge and compass: draw a circle of radius s centered at one vertex, then mark the two points on the circle that are distance s from that vertex; connecting the three points gives the triangle.

History and mathematical context

The equilateral triangle predates formal geometry and appears in ancient art and architecture. In classical geometry it served as a primary example of a regular polygon: Euclidean constructions with compass and straightedge include the equilateral triangle as one of the simplest constructible polygons. Its symmetry and simple trigonometric ratios make it a standard object in proofs and exercises.

Uses and examples

Equilateral triangles are common in engineering and design because of their structural stability and efficient tiling properties. Regular triangular meshes are used in computer graphics, finite-element analysis, and geodesic domes. In nature, equilateral patterns appear in crystal lattices, honeycomb variants, and molecular structures where symmetry lowers energy states.

Distinctions and notable facts

While some sources use "equilateral" and "regular" interchangeably for triangles, the terms emphasize different aspects: equilateral stresses equal side lengths, equiangular stresses equal angles, and regular indicates both. An equilateral triangle is the only triangle that is simultaneously equiangular, equilateral, and regular. Its internal decomposition into 30°–60°–90° triangles provides a common entry point to basic trigonometry and exact radical expressions for lengths.