Suppose $a,b,c \in\mathbb N$, and the value of $c$ is known and fixed, while $a$ and $b$ are unknown and are both smaller than $c$. What is the total number of unique triangles possible with $a, b$ and $c$ as its sides?
2026-05-15 01:30:22.1778808622
On
What is the number of triangles with integer sides, given the length of the longest side?
4.9k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
2
There are 2 best solutions below
0
On
Depending on whether you consider $(a,b,c)=(3,4,5)$ to give a different triangle from $(a,b,c)=(4,3,5)$, you seem to be either asking for the number of elements in $$ \{\,(a,b)\in\mathbb N^2\mid 0<a< n \text{ and } 0<b, n \text{ and } a+b>n\,\} $$ or for the number of elements in $$ \{\,(a,b)\in\mathbb N^2\mid 0<a\leq b< n \text{ and } a+b>n\,\}. $$ Surely you can find the answers to these easy questions? Make a drawing of dots inside a square if you need to.
The formula you need is $$\left\lfloor\frac{(c-1)^2}{4}\right\rfloor$$
As noted here, it counts "the number of noncongruent integer-sided triangles with largest side $c$". See this article as well.