We note that if we inscribe the $n$-gon in a circle, then according to the inscribed angle theorem, the angle between two sides is $\frac{1}{2}$ times some $x-y$, where $x$ and $y$ are integer multiples of the arc measure of one side of the $n$-gon. Thus, the angle is equal to $\frac{1}{2}$ times an integer multiple of $\frac{360}{n}$, so $\frac{1}{2} \cdot k \cdot \frac{360}{n}=28$ for some integer $k$. Simplifying gives $7 n=45 k$, and since all $k$ are clearly attainable, the smallest possible value of $n$ is 45.