A hyperbolic plane $\widehat{H}$ of positive curvature is the projective model of the de Sitter plane. In article the ways of measurement of the figures areas of the plane $\widehat{H}$ are offered. The cyclic orthogonal coordinate systems are described. One family of coordinate curves in such systems form by concentric cycles (by hyperbolic cycles, elliptic cycles or oricycles). Other family of coordinate curves form by the axes of these cycles. The formulas for the calculation of the figures areas of the plane $\widehat{H}$ are received.