Proof of the formulas for the the area of a rectangle and volume of a rectangular prism How do we prove that the area of a rectangle and the volume of a rectangular prism are the product of the measure of their sides?
 A: The Lebesgue measure in $\mathbb R^n$ is by definition the translation invariant meaasure that assigns $1$ to the unit hypercube. It follows immediately that the volume of an axe-parallel rectangle etc. is given by the product of the one-dimensional measures of its sides; similarly for axe-parallel products of lower-dimensional measurable figures and a height (such as prisms). For rotated rectangles, it may be advisable to show more generally the connection between a linear transformations and the absolute value of the determinant.
A: You need to first define area if you want to prove something about it. Or take it as a primitive notion and "define" (abuse of language here) it via axioms. If you want a proof in the style of classical geometry (which is the way we usually learn it in school, but without the proofs --haha), it's much harder to check if it is rigorous than an analytic one. Anyway, this is how many people initially learned it:
I will give just a sketch for your first question. That's because I don't think I can do more than it right now.
Axioms:
1) The area of a unit square is 1;
2) Congruent polygons have equal areas;
3) If a simple polygon P is decomposed into a finite number of parts, then the area of P is the sum of the areas of the parts.
Theorem: The area of a rectangle is the product of the lengths of its sides.
We will divide it into 3 cases:
1) Both lengths are whole numbers;
2) Both lengths are rational;
3) At least one of the lengths is an irrational number (the Greeks discovered this can happen in geometry as a consequence of the Pythagorean theorem).
Proof:
In case 1, let's say the sides are a and b, we divide the sides into parts of unit length and draw the perpendiculars. Then we can prove the formula for the area of the rectangle if we can show that the rectangle is decomposed into ab unit squares.
In case 2, we need to find a new unit of length that in commensurable with our original unit of length, and that will take us to the proof in case 1.
In case 3, things get more complicated... You can just be happy with approximations or use the method of exhaustion (or the theory of limits) to show that the formula is exact.
References:


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*ProofWiki: https://proofwiki.org/wiki/Axiom:Area_Axioms and https://proofwiki.org/wiki/Area_of_Square
Notes: 


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*You probably can find these things in books of Plane Euclidean Geometry, but I don't have any such book, so I can't give you a good indication.

*You could have used different axioms, like postulating Cavalieri's principle or the idea of a moving segment (like in calculus, but of constant length).

*In classical geometry, proofs are too verbal for my taste.

