Exercise 12.1 asks to explain more explicitly why the dimension of the configuration space for a rigid body in Euclidean 3-space is 6. As specified it seems to me that the correct dimension is 5. Why? There are 3 dimensions needed to specify the location of the center of mass of the rigid body and only two spherical coordinates to specify the orientation of the body assuming that a canonical orientation for the body has been chosen. There is no need for a radial component since all that is being specified is the direction the body is pointing and no magnitude of any sort related to that direction - such as for example speed of rotation.

What am I missing?

What about the extra degree of freedom needed to account for rotation about the line through the origin?