An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties.

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.

An equilateral triangle is a triangle that has three equal sides. 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. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base.

The follow-up definition above may result in more precise properties. For example, since the perimeter of an isosceles triangle is the sum of its two legs and base, the equilateral triangle is formulated as three times its side. The internal angles of an equilateral triangle are equal, 60°. Because of these properties, the equilateral triangles are regular polygons. 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. 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 ${\displaystyle \mathrm {D} _{3}}$ of order six. Other properties are discussed below.

The area of an equilateral triangle with edge length ${\displaystyle a}$ is $${\displaystyle T={\frac {\sqrt {3}}{4}}a^{2}.}$$ The formula may be derived from the formula of an isosceles triangle by Pythagoras theorem: the altitude ${\displaystyle h}$ of a triangle is the square root of the difference of squares of a side and half of a base. Since the base and the legs are equal, the height is: $${\displaystyle h={\sqrt {a^{2}-{\frac {a^{2}}{4}}}}={\frac {\sqrt {3}}{2}}a.}$$ In general, the area of a triangle is half the product of its base and height. The formula for the area of an equilateral triangle can be obtained by substituting the altitude formula. 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. Because all of the angles of an equilateral triangle are 60°, the formula is as desired.^{[citation needed]}

A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral. That is, for perimeter ${\displaystyle p}$ and area ${\displaystyle T}$, the equality holds for the equilateral triangle:$${\displaystyle p^{2}=12{\sqrt {3}}T.}$$

The radius of the circumscribed circle is:$${\displaystyle R={\frac {a}{\sqrt {3}}},}$$ and the radius of the inscribed circle is half of the circumradius:$${\displaystyle r={\frac {\sqrt {3}}{6}}a.}$$

A theorem of Euler states that the distance ${\displaystyle t}$ between circumcenter and incenter is formulated as ${\displaystyle t^{2}=R(R-2r)}$. As a corollary of this, the equilateral triangle has the smallest ratio of the circumradius ${\displaystyle R}$ to the inradius ${\displaystyle r}$ of any triangle. That is:$${\displaystyle R\geq 2r.}$$