Equilateral Triangle Calculator

The Equilateral Triangle Calculator computes all geometric properties of an equilateral triangle from a single measurement: the side length. An equilateral triangle has all three sides equal and all three interior angles equal to 60 degrees, making it the most symmetric triangle possible. Enter the side length to instantly obtain the area, perimeter, height, inradius, and circumradius.

Equilateral triangles have fascinated mathematicians, artists, and architects for millennia. In ancient Greece, the equilateral triangle was associated with harmony and balance. It appears in Euclid's Elements as the very first construction in Book I, Proposition 1: given a line segment, construct an equilateral triangle upon it using only a compass and straightedge. This foundational role underscores the triangle's importance in classical geometry.

The equilateral triangle possesses remarkable symmetry properties. It has three lines of symmetry (each from a vertex to the midpoint of the opposite side) and rotational symmetry of order three (120° rotations map it onto itself). Its symmetry group, the dihedral group D₃, has six elements. This high degree of symmetry means that many geometric quantities — centroid, circumcenter, incenter, and orthocenter — all coincide at a single point, unlike in general triangles where these are four distinct points.

In architecture and engineering, equilateral triangles appear in trusses, geodesic domes, and triangulated grids. Buckminster Fuller's geodesic dome, one of the most efficient structural forms ever devised, is built from networks of equilateral and near-equilateral triangles. The triangle warning sign used worldwide is an equilateral triangle, chosen for its visual stability and recognizability.

In nature, equilateral triangular symmetry appears in crystal structures, molecular geometry (such as the BF₃ molecule), and even in the arrangement of some biological structures. The hexagonal tessellation — one of only three regular tessellations of the plane — is composed of equilateral triangles (six equilateral triangles form each hexagon).

Mathematically, the equilateral triangle maximizes the area among all triangles with a given perimeter, making it the "optimal" triangle in an isoperimetric sense. Its height, area, inradius, and circumradius are all expressible as simple multiples of the side length involving $$\sqrt{3}$$, a constant that pervades equilateral triangle geometry.

The inradius is the radius of the largest circle that fits inside the triangle (the incircle), and the circumradius is the radius of the circle that passes through all three vertices (the circumcircle). For an equilateral triangle, the circumradius is exactly twice the inradius — a beautiful and unique relationship.