Area

The Euclidean area axioms The polygon section is a good place to look at area in general - we have seen that area can be measured differently depending on the geometry. In Euclidean geometry, the area of a regular polygon is given by A = .5(a)(s)(n) where a is the apothem, s is the side length, and n is the number of sides.

On the other hand, in hyperbolic geometry, the area is given by π A = δ 180 where δ is the defect.

So area depends on the axiomatic system we’re in. Of course, we all know what area means in Euclidean geometry..it’s...area...right? Axiom time:

The Euclidean area axioms • Existence postulate: To each region M in the plane there corresponds a real number Area M ≥ 0, called its area.

• Dominance postulate: For regions M1 and M2 in a plane, if M1 ⊆ M2,thenAreaM1 ≤ Area M2.

• Postulate of additivity: For any two plane regions M1 and M2 such that Area (M1 ∩M2)=0, then Area (M1 ∪ M2)=AreaM1 +AreaM2.

• Congruence postulate: Congruent regions in a plane have equal area.

• Unit of measure: The area of the unit square is one.

• Cavalieri’s principle: If all the lines parallel to some fixed line that meet the plane regions M1 and M2 do so in line segments having equal lengths, whose endpoints lie in the boundaries of the two regions, then Area M1 =AreaM2.

In summary, the first one tells you that something called area exists (similar to existency of say, angle measure), postulates 2, 3, 4, and 6 tell you some of the properties it has, and the fifth one (unit of measure) is like a protractor postulate, or defining a metric: it’s the only one that tells you how we measure Euclidean area - in terms of squares.

This axiom cannot exist in hyperbolic geometry, where we don’t have squares!