It is not defined on the axes :(If it's aligned with the axes, this is easy. Say the circular face is on the x and y axes with the height on the z axis. Check the two-dimensional distance from the center of the cylinder to (x, y). If it's greater than the radius, it's outside; if less, continue. (If equal then continue, but note that if the second test passes it's actually on the border rather than being inside.) For the second, compare the bottom and top z-coordinates of the cylinder to z. If z is between the two it's inside; if equal to one of the two it's on the boundary; if outside the two it's outside.