Equilateral triangle
| Equilateral triangle | |
|---|---|
 An equilateral triangle. Curved segment lines denote the internal angle, all of which is 60°. Single slashes indicate all edges are equal in length. | |
| Type | isosceles triangle regular polygon simplex |
| Edges and vertices | 3 |
| Schläfli symbol | |
| Symmetry group | of order six |
| Area | a quarter of the square root of 3 × side2 |
| Internal angle (degrees) | π/3 (60°) |
| Perimeter | 3 × side |
An equilateral triangle is a triangle with three sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular triangle. It is a special case of an isosceles triangle by modern definition. Its internal angle is 60° in degree or π/3 in radian. Its perimeter is three times the side length. Its area is a quarter of the square root of three multiplied with its squared side length. Its appearance is unchanged by three reflection lines, each of which passing through a vertex to the midpoint of the opposite edge, and three rotations around its centroid (0°, 120°, 240°).
Applications of an equilateral triangle are in popular culture, architecture, and other science fields such as the resemblance of the trigonal planar molecular geometry in stereochemistry, ternary plots in several fields, Einthoven's triangle in electrocardiography, as a solution of the Thomson problem and of the celestial mechanics's three-body problem, the Laser Interferometer Space Antenna's three identical observatory of gravitational waves spacecraft, and triangulation concept in satellite geodesy.
The two tessellation's instances are the triangular tiling where six equilateral triangles surrounds a common vertex, and the sphinx tiling as a special case of the polyiamond. Classes of polyhedra with equilateral triangles as their faces are deltahedra and the family of uniform antiprisms. The interconnection between equilateral triangles and topics in number theory are the Padovan sequence and the triangular numbers. Fractal shapes that are generated by equilateral triangles iteratively are the Sierpiński triangle and the Koch snowflake. An equilateral triangle can have three arcs on its sides, a result of the Reuleaux triangle
Many theorems, inequalities, and problems involves equilateral triangles. For example, Napoleon's theorem stated that the centroid of three equilateral triangles arranged to form the sides of an arbitrary triangle form an equilateral triangle.