PROBLEM:

Express the area A of an equilateral triangle as a function of its side x.

SOLUTION:

From the formula of the area of a triangle, \displaystyle A=\frac{1}{2} \text{a}\text{b} \sin\left(\theta \right). Also, we know that the interior angle of an equilateral triangle is 60 degrees, and \displaystyle \sin\:60^{\circ} =\frac{\sqrt{3}}{2}.

\begin{align*}
A & =\frac{1}{2} \text{a}\text{b} \sin\left(\theta \right) \\
A & =\frac{1}{2} \cdot x\cdot x\cdot \sin\:60^{\circ} \\
A & =\frac{1}{2}\cdot x^2\cdot \frac{\sqrt{3}}{2} \\
A & =\frac{\sqrt{3}}{4}x^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\
\end{align*}