# Length of the side of a discrete equilateral triangle from area

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.

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## 1 Answer

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.

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Looks promising, can you give me an example of this in practice? - I understand how to solve without the Coefficients of the x values. – MChandler Jun 14 '12 at 15:36
@MChandler: Ok. Suppose they gave you $55$. Then you are to solve $x^2 + x - 110 = 0$. This has solutions $\dfrac{-1 \pm \sqrt{1 + 440}}{2} = \dfrac{-1 \pm 21}{2} = -12, 10$. So that $n = 10$ in $n(n+1)/2$, and thus there are $10$ glasses to a side. – mixedmath Jun 14 '12 at 16:09
Okay, understanding this much better now, unsure as to how you got to the figure of 440 and why there is a ± before the √1+440. Many thanks for your help in advance, I also understand that this community isn't for giving lessons, but I can' seem to find much information on this on the internet - I think i'm just searching for the wrong things. – MChandler Jun 14 '12 at 17:54
@MChandler: Ah, I used the 'quadratic formula.' If you google it, it will tell you how I answered the quadratic. – mixedmath Jun 14 '12 at 21:01