Formula to figure out why the equilateral triangle area formula works

I know Prof. Loh asked for us to figure out why the equilateral triangle area formula works but I can't figure it out. Anyone have any ideas?

Also how does the little question at the end make sense?

This post is deleted! 
@TSSGraviser
I believe you're talking about the question with similar right triangles! Please correct me if I misunderstood.
Since it is a bit complicated to calculate the unknown side lengths of the bigger triangle with a hypotenuse of 314, we are using a simple triangle with a hypotenuse of 2. This simple triangle has side lengths of 1, \( \sqrt{3} \) and 2(hypotenuse), as explained by Prof. Loh in the previous example.
Between two similar triangles (triangles with same anglesin this case, 30°60°90°), corresponding sides are proportional to each other. For example, in this problem, the hypotenuse of the bigger triangle is 314 and that of the smaller triangle is 2, which means the sides are scaled up by a factor of \( \frac{314}{2} = 157 \). Therefore, for the unknown sides in the bigger triangle, you can multiply the side lengths of the smaller triangle by 157. As I mentioned, the side lengths of the smaller triangle are 1, \( \sqrt{3} \) and 2, so if you multiply each of these numbers by 157, you will get 157, 157\( \sqrt{3} \) and 314. This is how Prof. Loh got the side lengths of the bigger triangle for this question.
Happy Learning!
The Daily Challenge Team

@TSSGraviser
When you calculate the area of an equilateral triangle, you first need to know the height (altitude). In order to do so, you first take half of one side \( (\frac{s}{2}) \) and another side \( (s) \) and use the pythagorean theorem to solve for the height:
\( h = \sqrt{s^2(\frac{s}{2})^2} \)
\( h = \sqrt{s^2\frac{s^2}{4}} \)
\( h = \sqrt{\frac{4s^2}{4}\frac{s^2}{4}} \)
\( h = \sqrt{\frac{3s^2}{4}} \)
\( h = \frac{s}{2} \sqrt{3} \)The original formula for the area of a triangle is \( A = \frac{1}{2} (base)\times(height) \). In this case, \( base = s \) and \( height = \frac{s}{2} \sqrt{3} \). Now, let's calculate the area!
\( A = \frac{1}{2} (s)\cdot(\frac{s}{2} \sqrt{3}) \)
\( A = \frac{s^2}{4} \cdot \sqrt{3} \)
\( A = (\frac{s}{2})^2 \cdot \sqrt{3} \Rightarrow \) This is the equilateral triangle area formula!Thus, the equilateral triangle area formula comes from the original triangle area formula \( [A = \frac{1}{2} (base)\times(height)] \). This is why it works!
Happy Learning,
The Daily Challenge Team