Differential and Integral Calculus by Feliciano and Uy, Exercise 1.1 Problem 5

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


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

\displaystyle A=\frac{1}{2}\cdot x\cdot x\cdot sin\:60^{\circ}

\displaystyle A=\frac{1}{2}\cdot x^2\cdot \frac{\sqrt{3}}{2}

\displaystyle A=\frac{\sqrt{3}}{4}x^2