1. ## Area is unique.

In Euclidean Geometry we definie area as (for polygons) a positive real number satisfing:
1)Two congruent polygons have the same area.
2)The area of a rectangle is the product of base and height.
3)A polygon which is a union of two polygons not having an interior point is the sum of the areas of the two polygons.
$Prove:$The area of a polygon is well-defined.

Meaning no matter how you disect a polygon as smaller polygons and sum them you always end up the the same real number.

This problem seems extremely difficult. It seems to be connected to graph theory (get it? connected to graph theory ). How do you prove such a monster problem.
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However, if you define area not in terms of polygons but rather in terms of function. Then the problem is easier. You define it area to be the limit of the Riemann sum. And the fact is that no matter how a Riemann sum is partitioned as long as the norm is convergent to zero the limit (if exists) is unique. Thus, area is well-defined.

2. The definition of area for a surface, should go as follows:

Given a polygon, consider a (disjoint) partition of it into rectangles. We can find the area here but adding up the elements of the partition. The axiom guarantees this.

For a general surface, we can consider a disjoint covering of it, consisting of polygons. We can find the area of this covering, call it S. Then define the area of the surface as liminf{S}, where the limes inferior is considered over the set of all coverings of S.

This is well defined, and easily seen to comply with everything a measure of area should be Plus, it does not need anything more for its definition, but the area of rectangles!

3. My 10th Grade math teacher defined (and I still remember this), after he asked the class what area is, that area is the measure of a mathematical region. I just find it funny, he manages to bring Measure Theory into 10th grade.

4. Originally Posted by ThePerfectHacker
My 10th Grade math teacher defined (and I still remember this), after he asked the class what area is, that area is the measure of a mathematical region. I just find it funny, he manages to bring Measure Theory into 10th grade.
I have a problem with this definition: what does the term "measure" mean? I have the same problem with the formula d = rt (distance equals rate times time.) What the heck "rate" are we talking about? Not specific enough for me.

-Dan

5. Originally Posted by topsquark
I have a problem with this definition: what does the term "measure" mean? I have the same problem with the formula d = rt (distance equals rate times time.) What the heck "rate" are we talking about? Not specific enough for me.

-Dan
I think he meant the Lebesque Measure.

6. Originally Posted by ThePerfectHacker
I think he meant the Lebesque Measure.
Ah! That will do it.

-Dan