Finding the area of an equilateral triangle using the Pythagorean theorem

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?