We study general classes and properties of extremal and non-extremal static black-hole solutions of N = 2, d = 5 supergravity coupled to vector multiplets using the recently proposed H-FGK formalism, which we also extend to static black strings. We explain how to determine in general the integration constants and physical parameters of the black-hole and black-string solutions. We derive some model-independent statements, including the transformation of nonextremal flow equations to the form of those for the extremal flow. We apply our methods to the construction of example solutions (among others a new extremal string solution of heterotic string theory on K 3 × S 1 ) and analyze their properties. In all the cases studied the product of areas of the inner and outer horizon of a non-extremal solution coincides with the square of the moduli-independent area of the horizon of the extremal solution with the same charges. a