Distributions of distances between pairs of points within geometrical objects, or the closely related proximity functions and geometric reduction factors, have applications to dosimetric and microdosimetric calculations. For convex bodies these functions are linked to the chord-length distributions that result from random intersections by straight lines. A synopsis of the most important relations is given. The proximity functions and related functions are derived for right cylinders with arbitrary cross sections. The solution utilizes the fact that the squares of the distances between two random points are sums of independently distributed squares of distances parallel and perpendicular to the axis of the cylinder. Analogous formulas are derived for the proximity functions or geometric reduction factors for a cylinder relative to a point. This requires only a minor modification of the solution.