The hyperbolic plane has many properties of the usual Euclidean plane. However, since Euclid’s fifth postulate does not hold, it also has many strange features, different from Euclidean geometry. We study properties of the hyperbolic plane using the upper half-plane model \(\mathbb{H}\). Our main goal was to introduce the hyperbolic metric in a natural way. We consider the group of incidence-preserving Möbius transformations and study transitivity properties and conformality of the general Möbius group. After that we discover its subgroup preserving \(\mathbb{H}\). We use complex analysis to compute arc-length and path integrals, and obtain formulae for calculating hyperbolic length and distance. We discover some other metric properties of the hyperbolic plane. Besides the upper half-plane model, we also mention the Poincaré disk model and a general construction using complex analysis.