On the Perimeter and Area of the Unit Disc Juan Carlos Alvarez Paiva and Anthony Thompson Summary: On the Perimeter and Area of the Unit Disc Juan Carlos ´Alvarez Paiva and Anthony Thompson 1. INTRODUCTION. In the preamble to his fourth problem (presented at the International Mathematical Congress in Paris in 1900) Hilbert sug- gested a thorough examination of geometries that "stand next to Euclidean geometry" in the sense that they satisfy all the axioms of Euclidean ge- ometry except one. In non-Euclidean geometries the axiom that is usually taken to fail is the famous parallel postulate. This leads to the relatively well-known hyperbolic and elliptic geometries. The significance of these is that, like Euclidean geometry, they are homogeneous (all points have the same status) and isotropic (all directions have the same status). Another type of geometry that "stands next to Euclidean geometry" is the geometry of normed spaces. Here translating a line segment does not change its length, but the axiom that states that two triangles with equal corresponding sides are congruent no longer holds. These geometries are homogeneous but not isotropic. In this article we survey some of the most basic results on the geometry of unit discs in two-dimensional normed spaces, while adding a few results and some new proofs of our own. These results answer simple questions about the perimeter of the unit disc, its area, and the relationships between these Collections: Mathematics