Pick's Theorem

Given a lattice (a grid with equally disposed points), the following method can be used to measure the area of any given whole polygon.

$\begin{tikzpicture} % grid \draw[help lines,step=1] (-2,-2) grid (2,2); % coordinates outer \coordinate[label={[black]left:$A$}] (A) at (-1,1); \coordinate[label={[black]below:$B$}] (B) at (-1,-1); \coordinate[label={[black]right:$C$}] (C) at (1,-1); \coordinate[label={[black]right:$D$}] (D) at (2,2); % coordinate inner \coordinate[label={[black]right:$E$}] (E) at (0,0); \coordinate[label={[black]right:$F$}] (F) at (1,0); \coordinate[label={[black]right:$G$}] (G) at (1,1); \coordinate[label={[black]right:$H$}] (H) at (0,1); % polygon \draw[black, line width=.1mm] (A) -- (B) -- (C) -- (D) -- cycle; % outer lattice points \drawpoint{A}{1mm}{green} \drawpoint{B}{1mm}{green} \drawpoint{C}{1mm}{green} \drawpoint{D}{1mm}{green} % inner lattices points \drawpoint{E}{1mm}{red} \drawpoint{F}{1mm}{red} \drawpoint{G}{1mm}{red} \drawpoint{H}{1mm}{red} \end{tikzpicture}$

$\begin{eqnarray*} A_{\blacksquare} &=& {\color{green}i}+\frac{{\color{red}b}}{2}-1 \end{eqnarray*}$

Index