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