smc

Let $n$ be the number of sides in the polygon. The number of interior angles in the polygon is $180(n-2)$. We know that the sum of all but one of them is $2190^{\circ }$, so the sum of all the angles is more than that. $$180(n-2)>2190$$ $$n-2>12\frac{1}{6}$$ $$n>14\frac{1}{6}$$ The sum of the angles in a 15-sided polygon is $2340^{\circ }$, making the remaining angle $150^{\circ }$. The angles of a convex polygon are all less than $180^{\circ }$, and since adding one more side means adding $180^{\circ }$ to the measure of the remaining angle, we can confirm that there are $\boxed{15}$ sides in the polygon.