This paper is a synopsis of the two least squares approaches developed in  for the perpendicular joining of a flat graphene sheet with a carbon nanotube. The two least squares approaches are the variation in the bond length and the variation in the bond angle. These are used to examine the joined structure of a zigzag (8,0) carbon nanotube with a flat graphene sheet. There are sixteen possible distinct defects corresponding to the number of atoms at the (8,0) tube open end, and therefore, in total sixteen joining structures need to be investigated. Moreover, the polygons that occur at the junction are determined and are shown to be consistent with Euler’s theorem. Assuming that only pentagons, hexagons and heptagons are acceptable, the number of possible structures is greatly reduced, but there is only one structure that is physically meaningful. These purely geometrical approaches can be formally related directly to a certain numerical energy minimization method used by a number of authors [2-5].