Equilateral Triangle
GeometryAn 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. All three angles are equal, each measuring $60^\circ$. It is perfectly symmetrical in every direction.
Example
A yield sign and a Triforce symbol are equilateral triangles. If each side is $5$ cm long, all three sides are $5$ cm and all three angles are exactly $60^\circ$.
Key Insight
Because an equilateral triangle has three lines of symmetry, it looks the same from all three corners. It is the only triangle that is also a regular polygon - equal sides AND equal angles.
Definition
An equilateral triangle has three congruent sides and three congruent angles of $60^\circ$ each. It is both equilateral (equal sides) and equiangular (equal angles). For a side length $s$, the area is $(\sqrt{3}/4)s^2$ and the height is $(\sqrt{3}/2)s$. An equilateral triangle is a regular polygon with three sides.
Example
For an equilateral triangle with side $6$: height $= (\sqrt{3}/2)(6) = 3\sqrt{3}$ approximately $5.2$. Area $= (\sqrt{3}/4)(36) = 9\sqrt{3}$ approximately $15.6$. Perimeter $= 18$.
Key Insight
An equilateral triangle has three-fold rotational symmetry ($120^\circ$ rotations) and three lines of reflective symmetry. It tiles the plane (you can cover a flat surface with equilateral triangles without gaps), and this tiling property makes it important in crystallography and design.
Definition
An equilateral triangle with side $s$ has circumradius $R = s/\sqrt{3}$ and inradius $r = s/(2\sqrt{3})$. It is the unique triangle (up to similarity) achieving maximum area for a given perimeter (isoperimetric inequality for triangles). 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.
Example
For side $s = 2$: $R = 2/\sqrt{3} = 2\sqrt{3}/3$. $r = 1/\sqrt{3} = \sqrt{3}/3$. Note $R = 2r$, which is a property of equilateral triangles only.
Key Insight
The equilateral triangle is the 2D analog of the regular tetrahedron (3D). Both maximize symmetry in their dimension. The equilateral triangle's connection to the cube roots of unity via complex numbers shows how geometric symmetry corresponds to algebraic structure - a theme throughout abstract algebra (symmetry groups of regular polygons are cyclic or dihedral groups).