Firstly I haven't practised any mathematics in a long time, I understand that this might be pretty basic for math.stackexhcange, but I cannot seem to find any answers on the internet anywhere!
I've come across this problem at work, where basically if you are given $X$ amount of glasses to form into an equilateral triangle, how would you calculate the length of the sides - Using glasses as the unit of measurement?
I realise that you need to figure out if the number is triangular, and that there is only one answer for this problem as there is a standard scale for the sizes of the triangles - 3 glasses for the smallest Triangle, 6 for the next, 10, 15, 21 ect...
Formula for testing if the number is triangular is:
$$ (n/2) × (n + 1) $$
Then working backwards from the equation (assuming the number is triangular):
$$ \text{Area}=\text{Side}{^2} \frac{\sqrt3}{4} $$
And what I've been using to try to figure out the potental value of the side is:
$$ \text{Side}={\sqrt\frac{A}{(\frac{\sqrt3} 4)}} $$
I'm not sure if this formula works in this instance as what I'm measuring isn't using standard units of measurements, as there cannot be fractions of glasses. - All the results I've got from this are wrong.
If I understand you correctly, someone has made an 'equilateral triangle' out of some number $A$ of glasses, and you want to know how many glasses there are in the base. You realized the number must be triangular, so that $A = \dfrac{n(n+1)}{2}$.
Then I recommend that you solve the quadratic equation $2A = x(x+1) = x^2 + x$, i.e. $x^2 + x - 2A = 0$.There will be two solutions, and the positive one will give you the number of glasses to a side.