# What is area of equilateral triangle?

Jun 15, 2015

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:

$\sqrt{{a}^{2} - {\left(\frac{a}{2}\right)}^{2}} = \sqrt{{a}^{2} - {a}^{2} / 4} = \sqrt{\frac{3 {a}^{2}}{4}} = \frac{\sqrt{3} a}{2}$

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:

$\frac{\sqrt{3} a}{2} \cdot \frac{a}{2} = \frac{\sqrt{3}}{4} {a}^{2}$