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


fuss/mathematics/geometry/shapes.txt ยท Last modified: 2017/02/22 18:30 (external edit)

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