The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism.

It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry.

It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides.

[3][4] The internal angle of an equilateral triangle are equal, 60°.

The cevians of an equilateral triangle are all equal in length, resulting in the median and angle bisector being equal in length, considering those lines as their altitude depending on the base's choice.

[5] When the equilateral triangle is flipped across its altitude or rotated around its center for one-third of a full turn, its appearance is unchanged; it has the symmetry of a dihedral group

In general, the area of a triangle is half the product of its base and height.

[7] Another way to prove the area of an equilateral triangle is by using the trigonometric function.

The area of a triangle is formulated as the half product of base and height and the sine of an angle.

As a corollary of this, the equilateral triangle has the smallest ratio of the circumradius

is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem.

that can be packed into the equilateral triangle, but the open conjectures expand to

[11] Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.

The triangle of the largest area of all those inscribed in a given circle is equilateral, and the triangle of the smallest area of all those circumscribed around a given circle is also equilateral.

in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides is greater than or equal to 2, equality holding when

The equilateral triangle can be constructed in different ways by using circles.

An equilateral triangle can be constructed by taking the two centers of the circles and the points of intersection.

[17] An alternative way to construct an equilateral triangle is by using Fermat prime.

A regular polygon is constructible by compass and straightedge if and only if the odd prime factors of its number of sides are distinct Fermat primes.

Notably, the equilateral triangle tiles the Euclidean plane with six triangles meeting at a vertex; the dual of this tessellation is the hexagonal tiling.

A polyhedron whose faces are all equilateral triangles is called a deltahedron.

There are eight strictly convex deltahedra: three of the five Platonic solids (regular tetrahedron, regular octahedron, and regular icosahedron) and five of the 92 Johnson solids (triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, and gyroelongated square bipyramid).

[21] More generally, all Johnson solids have equilateral triangles among their faces, though most also have other other regular polygons.

[22] The antiprisms are a family of polyhedra incorporating a band of alternating triangles.

When the antiprism is uniform, its bases are regular and all triangular faces are equilateral.

[23] As a generalization, the equilateral triangle belongs to the infinite family of

[24] Equilateral triangles have frequently appeared in man-made constructions and in popular culture.

In architecture, an example can be seen in the cross-section of the Gateway Arch and the surface of the Vegreville egg.

charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the best solution known for

places the points at the vertices of an equilateral triangle, inscribed in the sphere.