<p>In this paper a new contractive type mapping known as mapping contracting area of a right-angled triangle or a rectangle is introduced. It is possible to reduce the area of a rectangle or a right-angled triangle with the help of such mappings. This is a geometric technique that is connected to the study of geometric characteristics of certain curves defined over metric spaces. Some suitable supporting examples of our proven results are given to show that our defined mapping is different from usual contractive type mappings. Two geometric figures are illustrated to explain the speciality of our mapping. Moreover, some applications of our fixed point theorem are given to the existence and Ulam–Hyers stability of solution of a sixth-order ordinary differential equation together with boundary conditions and to the fixed circle problem where the uniqueness of fixed point for a mapping is not necessarily needed.</p>