Determine the number of convex polygons all of whose sides are the square roots of positive integers which can be inscribed in a unit circle. Polygons that are rotations or reflections of each other are considered the same.
Any chord in a unit circle has length at most 2 (if it is a diameter), so all side lengths must be $1, \sqrt{2}, \sqrt{3}$, or 2 . We can do casework based on the number of sides:

- Triangles: The possible combinations of side lengths are $(\sqrt{2}, \sqrt{2}, 2),(1,1, \sqrt{3})$, $(1, \sqrt{3}, 2)$, and $(\sqrt{3}, \sqrt{3}, \sqrt{3})$. This is 4 possibilities.
- Quadrilaterals: The possible ordered combinations of side lengths are $(1,1,1,2)$, $(1, \sqrt{3}, \sqrt{2}, \sqrt{2}),(1, \sqrt{3}, 1, \sqrt{3}),(1, \sqrt{3}, \sqrt{3}, 1),(\sqrt{2}, \sqrt{2}, \sqrt{2}, \sqrt{2})$, and $(1, \sqrt{2}, \sqrt{3}, \sqrt{2})$. This is 6 possibilities.
- Pentagons: The possible combinations of side lengths are $(1,1,1,1,1, \sqrt{3})$, $(1,1,1, \sqrt{2}, \sqrt{2})$, and $(1,1, \sqrt{2}, 1, \sqrt{2})$. This is 3 possibilities.
- Hexagons: The only option is that all sides are 1 , so there is 1 possibility.

Putting these together, there are $4+6+3+1=14$ polygons total.