Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.