Take any two points, the line segment conecting them lies either entirly within R or it meets the inner boundary at two points. In the latter case connect the two ponits on the inner boundary by the minor arc.
Thus either two points are connected by the line segment between them if this is within the annulus or by a pair of line segments and an arc otherwise.
Alternativly take the two points as , in polars, then and .
Then the curve:
lies entirly within the annulus and conects the two points
Modifying these to apply to any annulus I leave to you.