# Definitions Wikipedia, defines $\pi$ as the circumference of the circle with diameter $1$. Note here that because the circumference of a circle is directly proportional to its diameter (circles are similar figures), the circumference of circle of diameter $d$ is $\pi d$. It is also well know that $\pi$ is the area of the unit circle (circle of radius $1$). Maybe Wikipedia uses the first definition because it is more elementary to define and compute a line integral. However, the equivalence of both definitions follows from the [[Divergence theorem|divergence theorem]]. Recall that the divergence theorem yields that for an open set $V \subset \mathbb{R}^{2}$, a vector field $X\in C^{2}(V, \mathbb{R}^{2})$, and for a bounded set $\Omega \subset V$ with a smooth boundary, we have that $\int_{\Omega} \text{div}X = \int_{\partial \Omega} X \cdot \nu \, dS \tag{1}$where $\nu:\partial \Omega \to \mathbb{R}^2$ is the outward pointing normal vector. Now let $X=\text{Id}_{\mathbb{R}^{2}}$ so that $\text{div}(X)=2$ and $\Omega$ be the unit circle. Then $\nu(x,y)=(x,y)$ which yields $X(x,y) \cdot \nu(x,y)=x^{2}+y^2=1$. Now since $\Omega$ has diameter $2$, its circumference is given by $\int_{\partial \Omega} dS = 2\pi$ so $(1)$ gives the area of $\Omega$ is $\pi$. # Properties