# Thread: Find number of different triangles possible?

1. ## Find number of different triangles possible?

The sides of a triangle are a, b and c where a, b and c are integers and $a\leq b\leq c$. If c is given, show that the number of different triangles is $\frac{1}{4}c(c + 2)$ or $\frac{1}{4}(c + 1)^2$, according as c is even or odd. Also, show that the number of isoceles or equilateral triangle is $\frac{1}{2}(3c - 2)$ or $\frac{1}{2}(3c - 1)$, according as $c$ is even or odd.

2. The important fact here is that in a triangle with sides $a,b,c$, it must be true that $a+b>c, a+c>b,$ and $b+c>a.$

The last two inequalities are obvious given that $a\leq b\leq c$, so we just need to see how many ways we can choose $a$ and $b$ so that $a+b>c.$

Suppose $c$ is even.

Then if $a=1$, then $b$ must equal $c$.
if $a=2$, then $b$ can equal $c$ or $c-1$.
..
if $a=c/2$, then b can equal $c/2+1, ... , c$.

(So far we have $1+2+\cdots+c/2=\frac{1}{8}c(c+2)$ ways to choose $a$ and $b$.

Doing the same thing when $a>c/2$, we find that there are $\frac{1}{8}n(n+2)$ more ways to choose, giving a total of $\frac{1}{4}c(c+2)$ triangles.

The other cases of this problem are not much different.