পুনরালোচনা: ত্রিভুজের জন্য "জুতার ফিতা" ফর্মুলা

According to the Shoelace Theorem for a convex polygon with three coordinates (namely a triangle), with coordinates $(a, r)$, $(b, s)$, and $(c, t)$, we list the coordinates vertically and diagonally multiply.

$\rightarrow$ $(a,r)$

$\rightarrow$ $(b,s)$

$\rightarrow$ $(c,t)$

The area is $$\dfrac {\left| a \cdot s + b \cdot t + c \cdot r - r \cdot b - s \cdot s - t \cdot a \right|}{2}$$ Hopefully you see how we form a wrap-around shoelace!

Note that the triangle is only a special case of the Shoelace Theorem. The Shoelace Theorem applies in general for any convex polygon.

Suppose the polygon $P$ has vertices $(a_1, b_1)$, $(a_2, b_2)$, ... , $(a_n, b_n)$, listed in clockwise order. Then the area of $P$ is

$$\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1)|$$

The Shoelace Theorem gets its name because if one lists the coordinates in a column, $$\begin{align*} (a_1 &, b_1) \\ (a_2 &, b_2) \\ & \vdots \\ (a_n &, b_n) \\ (a_1 &, b_1) \\ \end{align*},$$ and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes. (AoPS Wiki)

Note that if you are trying to find the area of a parallelogram $ABCD$ with the shoelace theorem, it suffices to find the area of one triangle (say $\triangle ABC$) and then double it. This is because drawing a diagonal creates two congruent triangles.