The density functional theory (DFT), semiempirical molecular orbital calculations, and pi-orbital axis vector (POAV) analysis are used to study the energetics and geometries of carbon nanoconic tips. The tip models are formed by introducing two to five pentagons into a graphene network corresponding to experimentally observed cone angles of 19 to 84 degrees. The calculations show a pronounced energetic preference of (1,1) pentagons over (2,0) pentagons in tip structures, where one pentagon is at hexagonal coordinate (1,1) or (2,0) relative to the other at (0,0). The DFT energies of tips containing (1,1) pentagon pairs are lower than those with (2,0) pairs by 18.0-55.9 kcal/mol. This is attributed to less in-plane and out-of-plane strain being induced by (1,1) pentagon pairs than (2,0) pentagon pairs. The bond lengths, bond singles, torsion angles, and POAV angles are presented and discussed. The isolated pentagon rule is also tested for tip molecules containing two pentagons. The lower tip energy is obtained by increasing (a,a) pentagon separation or by decreasing (b,0) pentagon separation with b > 1.