Question

What is area of equilateral triangle?

Answer 1

If the sides of an equilateral triangle are all of length $a$, then the area is $\frac{\sqrt{3}}{4}{a}^2$.
Explanation:
Consider an equilateral triangle with sides of length $a$.
If you bisect it to make two right angled triangles, then those triangles will have hypotenuse of length $a$, shortest side of length $\frac{a}{2}$ and other side of length:
$\sqrt{a^2-{\left(\frac{a}{2}\right)}^2}=\sqrt{a^2-\frac{a^2}{4}}=\sqrt{\frac{3a^2}{4}}=\frac{\sqrt{3}a}{2}$
The two right angled triangles can be rearranged (turning one over) into a rectangle with sides $\frac{\sqrt{3}a}{2}$ and $\frac{a}{2}$.
The area of the rectangle, which is the same as the area of the original triangle is:
$\frac{\sqrt{3}a}{2}.\frac{a}{2}=\frac{\sqrt{3}}{4}{a}^2$