Radiative heat transfer in a complex two-dimensional enclosure with an absorbing, emitting, and isotropically scattering gray medium is investigated by using the finite-volume method. In particular, an implementation of the unstructured polygonal meshes, based on an unstructured triangular system, is introduced, thereby reducing the computation time required for the analysis of thermal radiation. By connecting each center point of the triangular meshes rather than joining the centroids of the triangular elements to the midpoints of the corresponding sides to form a polygonal element, it not only prevents concave shapes of adjacent faces, it also reduces the number of control-volume faces of the polygonal mesh. After a mathematical formulation and corresponding discretization equation for the radiative transfer equation are derived, the final discretization equation is introduced with the conventional finite-volume approaches. The present formulation is then validated by comparing results with those obtained in previous related works. All the results presented in this work show that the present method is accurate and computationally efficient for the analysis of radiative heat transfer problems in complex geometries.