The Uniqueness of Hyperbolic Area Theory

Any reasonable area function should have the following properties:

 
1. for every R;
2. if and intersect only in edges and vertices, then
   
   
3. if , then .

There is such a function--the defect. Recall . We know that satisfies (1), (2), and (3). In fact, we have the following theorem:

Theorem 17.9:    Let

be an area function satisfying (1), (2), and (3). Then there is a positive constant k>0 such that

for every .

We are not able to prove this at this time. The proof is within your grasp, but we do not have time to do it.

Corollary: In the area of any triangle is at most .

There is no finite triangle whose area equals the maximal value , although you can approach this area as closely as you wish (and achieve it with a trebly asymptotic triangle). J. Bolyai proved that you can construct a circle of area and a regular 4-sided polygon with a angle that also has this area.