For a positive integer n, an n-sided polygon lying on a circular arc or, shortly, an n-fan is a sequence of $$n+1$$ n+1 points on a circle going counterclockwise such that the “total rotation” $$\delta $$ δ from the first point to the last one is at most $$2\pi $$ 2π . We prove that for $$n\ge 3$$ n≥3 , the n-fan cannot be constructed with straightedge and compass in general from its central angle $$\delta $$ δ and its central distances, which are the distances of the edges from the center of the circle. Also, we prove that for each fixed $$\delta $$ δ in the interval $$(0, 2\pi ]$$ (0,2π] and for every $$n\ge 5$$ n≥5 , there exists a concreten-fan with central angle $$\delta $$ δ that is not constructible from its central distances and $$\delta $$ δ . The present paper generalizes some earlier results published by the second author and Á. Kunos on the particular cases $$\delta =2\pi $$ δ=2π and $$\delta =\pi $$ δ=π .