How can I check if a polygon can completely contain a circle of a certain radius?
This is likely more complicated than you would prefer, but: Compute the medial axis, which immediately yields the largest disks that fit inside the polygon:
their centers are vertices (degree $\ge 3$) of the axis (see the
figure below).


  Chin, Francis, Jack Snoeyink, and Cao An Wang. "Finding the medial axis of a simple polygon in linear time." Discrete &amp; Computational Geometry 21.3 (1999): 405-420.



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(Image from Alexander Tsvyashchenko.)




See also, Largest circle inside a non-convex polygon for an ad hoc approach
(whose correctness I did not attempt to verify).