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. Unknown error_msg)
credits: created by pickten --Pickten 18:22, 12 June 2011 (EDT)