# How to find the sides of an equilateral triangle given all angles.

How do I find the length of sides and the height of an equilateral triangle when I only know the three angles and the area. The area is 50.3144 and obviously all the angles are 60 degrees. I'm in grade 10 so I don't want some crazy formula to figure this out, I've looked everywhere on the internet but all websites talk about SAS, AAS etc. but I have not been given any side length just the area and angles.

• Remember I don't know any side lengths – megzy027 Apr 30 '16 at 10:05

Let the triangle be $\;\Delta ABC\;$, say with $\;A\;$ the upper vertex, and let $\;D\;$ be the midpoint of side $\;BC\;$ , say. Then, $\;\Delta ABD\;$ is a straight $\;30-60-90\;$ triangle , so $\;|BD|=\frac12|AB|\;$ and $\;|AD|=\frac{\sqrt3}2|AB|\;$ (use Pythagoras Theorem) , and thus the whole triangle's area is

$$\frac{|BC|\cdot |AD|}2=\frac{|AB|\cdot\frac{\sqrt3}2|AB|}2=\frac{\sqrt3}4|AB|^2\stackrel{\text{given}}=50.3144$$

So now just get (fill in details)

$$|AB|=\sqrt{\frac{4\cdot50.3144}{\sqrt3}}$$

Well there is simple formula to find the area of an equilateral triangle.

$Area=\frac{a^2\sqrt{3}}4$, where $a$ is the side. Now you do know the area. Compare it to this formula and and find $a$.

This formula can be easily proved in two simple ways that you might know. The first is using basic trigonometry, where you drop a perpendicular from a vertex to the opposite side and sum the areas of the two triangles that you get. And the second is using the semi-perimeter method. If you are unable to prove it, you could very well just remember the formula.

The area of any triangle is given by the length of the base times the height divided by 2. You are trying to calculate the length of the base.

If you draw your triangle you will easily see that if you bisect it vertically you have a right angle triangle with base $b/2$ and hypotenuse $b$. You can see by pythagoras that the height = $(b.\sqrt{3})/2$

You can then substitute the height and area into $a=b.h/2$ to find the base.

$a=b^2.\sqrt{3} /4$

$b=\sqrt{4a/\sqrt{3}}$

which is about 10.77