座標法
n個の頂点 (x1,y1),(x2,y2),...,(xn,yn) から成る自己交差を持たない多角形の面積は
S = 1 2 | ∑ k = 1 n ( x k y k + 1 − x k + 1 y k ) | = 1 2 | ∑ k = 1 n ( x k − x k + 1 ) ( y k + y k + 1 ) | = 1 2 | ∑ k = 1 n x k ( y k − 1 − y k + 1 ) | = 1 2 | ∑ k = 1 n y k ( x k − 1 − x k + 1 ) | {\displaystyle {\begin{aligned}S&={\frac {1}{2}}\left|\sum _{k=1}^{n}(x_{k}y_{k+1}-x_{k+1}y_{k})\right|\\&={\frac {1}{2}}\left|\sum _{k=1}^{n}(x_{k}-x_{k+1})(y_{k}+y_{k+1})\right|\\&={\frac {1}{2}}\left|\sum _{k=1}^{n}x_{k}(y_{k-1}-y_{k+1})\right|\\&={\frac {1}{2}}\left|\sum _{k=1}^{n}y_{k}(x_{k-1}-x_{k+1})\right|\end{aligned}}}