Text defining length, area and volume

Question

I am looking for a geometry textbook that axiomatises concepts such as length, area and volume of objects in Euclidean space; for example, the surface area of a $2$-sphere in $3$-space. Such a text would then define length, area and volume, showing that the definitions satisfy the axioms, perhaps uniquely. It may even give different definitions that satisfy some common set of axioms. Is there such a book?

Answer 1

These concepts are motivated through a measure theoretic approach, not just for Euclidian space, but for more general regions as well. I would recommend researching the Lebesgue measure in a real analysis textbook.