 
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 nonEuclidean geometries the axiom that is usually
taken to fail is the famous parallel postulate. This leads to the relatively
wellknown 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 twodimensional 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
