From an equilateral triangle $T$ where each side have a length of $L$. What is the area of $T$?
According to the Wikipedia page of equilateral triangles, the area is $$A=\dfrac{\sqrt{3}}{4}L^2$$
I am trying to solve this problem by using the Pythagorean theorem, as explained in this question, I can split the triangle in half to try and get the height.
Using the Pythagorean theorem, $$L^2=(\dfrac{L}{2})^2 + H^2$$
I can then isolate $H$ with :
$$H=\sqrt{L^2-(\dfrac{L}{2})^2}$$
Using the $A=\dfrac{1}{2}bh$ formula. I could then conclude with : $$A=\dfrac{L\sqrt{L^2-(\dfrac{L}{2})^2}}{2}$$
As said previously, the Wikipedia page shows something very different. What went wrong?
With the Pythagorean theorem, you can find the altitude of an equilateral triangle by dropping a vertical line to split it. Then, the long side and half of the bottom are, respectively, the hypotenuse $C$ and a short leg $B$ of a right triangle. The vertical will be side $A$ of a right triangle where $A=\sqrt{C^2-\bigl{(}\frac{C}{2}\bigr{)}^2}=\sqrt{\frac{4C^2-C^2}{4}}=\frac{C\sqrt{3}}{2}$. The area of one triangle is $\frac{1}{2}A*B$ where $B=\frac{C}{2}$. However, we have two of these right triangles so the area is simply A*B. You should end up with $area=A\cdot B=\frac{C\sqrt{3}}{2}\cdot\frac{C}{2}=\frac{C^2\sqrt{3}}{4}$.