# 9.2.4. Polygon¶

## 9.2.4.1. Area and Barycenter¶

Polygon area is determined by

\begin{split}\begin{align} \mathbf{A} &= \frac{1}{2} \sum_{i=0}^{n-1} P_i \otimes P_{i+1} \\ &= \frac{1}{2} \sum_{i=0}^{n-1} \begin{vmatrix} x_i & x_{i+1} \\ y_i & y_{i+1} \end{vmatrix} \\ &= \frac{1}{2} \sum_{i=0}^{n-1} x_i y_{i+1} - x_{i+1} y_i \end{align}\end{split}

where $$x_n = x_0$$

Polygon barycenter is determined by

\begin{split}\begin{align} \mathbf{C} &= \frac{1}{6\mathbf{A}} \sum_{i=0}^{n-1} (P_i + P_{i+1}) \times (P_i \otimes P_{i+1}) \\ &= \frac{1}{6\mathbf{A}} \sum_{i=0}^{n-1} \begin{pmatrix} (x_i + x_{i+1}) (x_i y_{i+1} - x_{i+1} y_i) \\ (y_i + y_{i+1}) (x_i y_{i+1} - x_{i+1} y_i) \end{pmatrix} \end{align}\end{split}

References

## 9.2.4.2. Moments of Inertia¶

Warning

untrusted formulae

\begin{split}\begin{align} I_x &= \frac{1}{12} \sum (y_i^2 + y_i y_{i+1} + y_{i+1}^2) (x_i y_{i+1} - x_{i+1} y_i) \\ I_y &= \frac{1}{12} \sum (x_i^2 + x_i x_{i+1} + x_{i+1}^2) (x_i y_{i+1} - x_{i+1} y_i) \\ I_{xy} &= \frac{1}{24} \sum (x_i y_{i+1} + 2 x_i y_i + 2 x_{i+1} y_{i+1} + x_{i+1} y_i) (x_i y_{i+1} - x_{i+1} y_i) \end{align}\end{split}

Reference