# Equilateral Triangle An **equilateral triangle** is a [[triangle]] in which all three sides have the same length. As a consequence, all interior angles are the same ($60^\circ$). ## Area and Height of an Equilateral Triangle +-- { .image} [[EquilateralTriangleLabelled.png:pic]] =-- Consider an equilateral triangle with side length $1$ and let $h$ represent its height. The area is then $\frac{1}{2} h$. More generally, if an equilateral triangle has base length $x$ then its height is $h x$ and so its area is $\frac{1}{2} h x^2$. Cutting the triangle in half from a vertex to the midpoint of the opposite side results in two triangles that can be put back together to form an [[isosceles]] triangle with angles $120^\circ$, $30^\circ$, and $30^\circ$. The base of this triangle is twice the height of the original, so is $2 h$. +-- { .image} [[EquilateralTriangleDissected.png:pic]] =-- Three of these triangles fit together to form a new equilateral triangle. As this new triangle has base $2 h$ its area is $\frac{1}{2} h (2 h)^2 = 2 h^3$. It also is three copies of the original triangle so its area is $\frac{3}{2} h$. Putting these together, $h^2 = \frac{3}{4}$ and so $h = \frac{\sqrt{3}}{2}$. The height and area of an equilateral triangle of base length $1$ are therefore respectively $$ \frac{\sqrt{3}}{2}, \qquad \frac{\sqrt{3}}{4} $$ and of base length $x$ $$ \frac{\sqrt{3}}{2} x, \qquad \frac{\sqrt{3}}{4} x^2 $$ [[!redirects equilateral triangles]] [[!redirects equilateral]] [[!redirects height of an equilateral triangle]] [[!redirects area of an equilateral triangle]] [[!redirects lengths in equilateral triangles]] [[!redirects lengths in an equilateral triangle]] [[!redirects angles in equilateral triangles]] [[!redirects angles in an equilateral triangle]] category: triangles