This paper is a self-independent continuation of my article at the previous edition of ICon-MaSTEd Conference, 2020, in which I dealt with the project Comparative geometry between the plane and the sphere. Below I discuss the possibility of adding a third kind of geometry to the plane and the sphere, namely, the hyperbolic geometry on the hemisphere. I describe my own path to the topic, then a possible structure of the syllabus containing basic concepts of hyperbolic geometry for future preschool and primary school teachers, and give arguments for introducing the topic in primary and secondary general education, not only for the “talented”, but also for the “average” students.