Конспект: Формула площі Гауса для многокутника

The shoelace theorem states that if a polygon has consecutive vertices $(a_1,b_1), (a_2,b_2), (a_3,b_3), ...(a_n, b_n)$, then the area of the polygon is given by $0.5 \cdot |D(a_1,a_2,b_1,b_2) + D(a_2,a_3,b_2,b_3)+D(a_3,a_4,b_3,b_4)+...+D(a_{n-1},a_n,b_{n-1},b_n)+D(a_n,a_1,b_n,b_1)|$

Notice that there's a big absolute value around the thing.

The function $D(w,x,y,z)$ is defined to be $wz-xy$. We denote it as D because it's the determinant of the matrix

$$\left[\begin{array}{lr} a_1 & b_1 \\ a_2 & b_2 \end{array}\right]$$

According to the Shoelace Theorem for a general polygon with vertices $(x_1,y_1)$, $(x_2,y_2)$, $\cdots$, $(x_n,y_n)$, the area of the polygon is $$A = \frac {\left| \displaystyle\sum_{i=1}^{n}\left( x_i y_{i+1} - y_1 x_{i+1}\right) \right|}{2},$$where $x_{n+1}=x_1$ and $y_{n+1}=y_1$.

Why is this called the shoelace theorem?

If you write the coordinates in the counterclockwise order as a list and you trace the path of the product sum, you should find that the path makes the shape of a wrap-around shoelace pattern! Here's a thought question to finish the topic.

Since the entire sum of products would just change sign but there is an absolute value, $A$ would remain the same.