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

]]>There's no way to further simplify \( 150 \sqrt{3} \), so you would leave it as is.

Happy Learning,

The Daily Challenge Team

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