In this diploma thesis, we show how to calculate the ark length of a plane curve in the planar p-metric. We first give some basic concrete examples of metrics and then continue with presenting the standard distance functions in Euclidian spaces. We get familiar with the taxicab metric, maximum metric and the more general p-metric. We show that each of them satisfies the four axioms of metric functions and present some basic properties of p-metrics.
In the second and main part of our diploma thesis, we show how to calculate the arc length of a curve in all these different metrics in the plane, using the geodesic approximation method.