The paper refers to isoperimetric problems for polygonal curves on surfaces of constant curvature. The authors introduce “hyperbolic” and “elliptic” areas and lengths that are based on non-Euclidean Heron’s formulas, and they establish a unified Heron formula for a triangle and a unified Brahmagupta formula for a cyclic quadrilateral on surfaces of constant curvature. Based on this, they give a unified isoperimetric inequality for polygonal curves in the Euclidean and hyperbolic plane as well as on the sphere. So they obtain many new (discrete analytical) inequalities involving trigonometric functions and hyperbolic trigonometric functions.