# Area of an equilateral triangle

The area of an equilateral triangle is , where is the sidelength of the triangle.

## Proof

*Method 1* Dropping the altitude of our triangle splits it into two triangles. By HL congruence, these are congruent, so the "short side" is .

Using the Pythagorean theorem, we get , where is the height of the triangle. Solving, . (note we could use 30-60-90 right triangles.)

We use the formula for the area of a triangle, (note is the length of a base), so the area is

*Method 2* **(warning: uses trig.)** The area of a triangle is . Plugging in and (the angle at each vertex, in radians), we get the area to be $\frac{s\cdot s\cdot \frac{\sqrt{3}{2}}{2}=$ (Error compiling LaTeX. ! File ended while scanning use of \frac .)