What does area represent?

Question

Since any two Euclidean shapes have an infinite number of points inside of them, shapes with different area have the same infinite number of points in them (and any object has the same number of points in it as are inside the plane it is inside). So area isn't a measure of the amount of points in an object, right? And if this is true, what does the area of a shape actually represent? In other words beyond the abstract idea of the amount of space in an object, what does an area of, for example, $4$, mean?

Answer 1

Area (in $\mathbb R^2$) starts with agreeing that the area of a rectangle of sides $a$ and $b$ has area equal to $a\cdot b$. You may take that as an axiom. Now, the area of more complicated shapes is obtained by approximating the shape by rectangles. This can be done in different ways and there are plenty of things to prove here, as well as some surprises. For instance, one can approximate a given shape from the outside: cover it by countably many disjoint rectangles, and add their areas. Take the infimum over all such covers, and that will give you a measure of the area of the shape measured from the outside. Similarly, you can inscribe rectangles inside the shape, add their areas, and take the supremum over all such numbers. This will give you a measure of the area of the shape measured from the inside.

Your question is a prelude to measure theory, a very important part of modern analysis. One of the important surprises is that the concept of area (and even the concept of length) can't be extended sensibly to measure all subsets of the plane (line). A famous example is Vitali's Set. So, the resulting theory is quite subtle.

Answer 2

It tells you that you could cover the region with 4 $1\times 1$ squares. If it's a weird shape, it would be more informative to say that if you cut it out of material and a $1\times 1$ square of that material weighs 1 ounce, then the shape you cut out would weigh 4 ounces.

Answer 3

Yes, area isn't a measure of the amount of points of an object. A positive measure P of an object J implies that J can get further subdivided into parts, since P consists of a positive real number and for all positive numbers, there exists a real number X such (X+X)=P. A point T consists of something which has no part, making subdivision into parts impossible and thus T has measure of zero.

The area of a shape represents how much space a 2-dimensional object would take up in a 2-dimensional plane.