Contents

# Contents

## Idea

The volume of a surface.

## Definition

### Of polygons

Let $\mathrm{Polygons}$ be the set of all polygons in the Euclidean plane $\mathbb{R}^2$. Then the area is a function $A:\mathrm{Polygons} \to \mathbb{R}$ such that for all polygons $P \in \mathrm{Polygons}$,

• $A$ is invariant under translations:

• Given a linear transformation $L$ and a polygon $P$, $A(L P) = \det(L) A(P)$

• Given two vertices $p$ and $q$ of $P$, …

### In terms of Jordan content

Given a large set $M$ of Jordan-measurable subsets of $\mathbb{R}^2$ bounded by a Jordan curve called shapes, the area of a shape $S \in M$ is the Jordan content of $S$.