%\bigskip
%\noindent\textbf{\large Polygon area}
\section{Polygon area}

For an arbitrary polygon with $N$ vertices, $(x_i, y_i)$, 
\begin{displaymath}
  A = 
       \frac{1}{2}\left|
         \begin{array}{cc}
           x_{N-1}  & x_0 \\
           y_{N-1}  & y_0 
         \end{array}
         \right|
         + 
      \frac{1}{2}\sum_{i = 0}^{N - 1} 
            \left|
              \begin{array}{cc}
                x_i &  x_{i + 1} \\
                y_i &  y_{i + 1}
              \end{array}
            \right|
\end{displaymath}
This is a signed area, positive if you go counter-clockwise around
the polygon, negative if you go clockwise.  The first term is
needed to cover the last section of the polygon.

\bigskip
Formula for the area of a circular segment (shaded):

\includegraphics[height=1in]{images/Chord}
$$A = \frac{1}{2} R^2 \cos^{-1} \left( {\frac{r}{R}} \right) - r \sqrt{R^2-r^2}$$

This can also be computed by subtracting the area of the triangle (origin to each chord endpoint) from the area of the sector (which is $\frac{R^2}{2} \theta$).

\smallskip
Surface area of a sphere is $4 \pi r^2$.
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