Equilateral Triangle
An equilateral triangle has all three sides equal in length and all three angles equal to 60 degrees.
Formula
\text{Area} = (\sqrt{3}/4)s^2; \text{all angles} = 60^\circ
Definition
An equilateral triangle has all three sides the same length and all three angles equal, each measuring $60^\circ$, perfectly symmetrical in every direction; it is both equilateral (equal sides) and equiangular (equal angles), making it a regular polygon with three sides. For side length $s$: area $= (\sqrt{3}/4)s^2$, height $= (\sqrt{3}/2)s$, circumradius $R = s/\sqrt{3}$, and inradius $r = s/(2\sqrt{3})$, so $R = 2r$, a property unique to equilateral triangles.
Example
A yield sign and a Triforce symbol are equilateral triangles: if each side is $5$ cm, all three angles are exactly $60^\circ$. For side $6$: height $=(\sqrt{3}/2)(6)=3\sqrt{3}\approx5.2$, area $=(\sqrt{3}/4)(36)=9\sqrt{3}\approx15.6$, perimeter $=18$. In the complex plane, the vertices of an equilateral triangle centered at the origin can be written as $\{r, r\omega, r\omega^2\}$, where $\omega = e^{2\pi i/3}$ is a primitive cube root of unity.
Key Insight
With three lines of symmetry, an equilateral triangle looks the same from all three corners, and it is the only triangle that is also a regular polygon; it also has three-fold rotational symmetry and tiles the plane without gaps, important in crystallography and design. It is the unique triangle (up to similarity) achieving maximum area for a given perimeter, and its connection to the cube roots of unity shows how geometric symmetry corresponds to algebraic structure, a theme running throughout abstract algebra, where symmetry groups of regular polygons are cyclic or dihedral.