Assuming you want an expression for the possible total area of the patio: if "completed" would be a rectangle of dimensions n, m: each must be >=3 to allow for octagonal corners. But each of the 4 corners must be missing, that's diagonals comprising (1/2)a^2, (1/2)b^2, (1/2)c^2, and (1/2)d^2, where a,b,c, and d must be variously limited and co-limited so as to allow at least 1 linear side of the original rectangle to be exposed. So A = (n*m)-(1/2)a^2-(1/2)b^2-(1/2)c^2-(1/2)d^2 as an expression. Now, imagine replicating your possible (potentially irregular) octagons onto a plane and juxtaposing them so as to create a paved network. What geometric properties might such a network have? You now have a miniscule idea what nature does with silicate networked minerals, except that takes place in 3-D, with tetrahedra of SiO4 .
