The area of a polygon and the shoelace formula

If the coordinates of the vertices of a polygon are known, its area can be computed by simply walking once around them¹. We do it for the pentagon with vertices $A(0, 0)$ , $B(4, 0)$ , $C(5, 3)$ , $D(2, 5)$ and $E(-1, 2)$ .

The shoelace formula

List the vertices counterclockwise as $(x_1, y_1), \ldots, (x_n, y_n)$ , form $x_i y_{i+1} - x_{i+1} y_i$ for each adjacent pair, and add. After the last vertex one returns to the first.

S = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right|

Working through it

Side	$x_i y_{i+1} - x_{i+1} y_i$	Value
$AB$	$0 \cdot 0 - 4 \cdot 0$	$0$
$BC$	$4 \cdot 3 - 5 \cdot 0$	$12$
$CD$	$5 \cdot 5 - 2 \cdot 3$	$19$
$DE$	$2 \cdot 2 - (-1) \cdot 5$	$9$
$EA$	$(-1) \cdot 0 - 0 \cdot 2$	$0$

The total is $40$ , so the area is $\dfrac{1}{2} \times 40 = 20$ .

What the formula means

The meaning can be traced back to the area of a triangle. The triangle formed by the origin and two points $(p, q)$ and $(r, s)$ has area $\dfrac{1}{2}|ps - qr|$ . Viewing each side from the origin cuts the figure into triangles; adding them with signs cancels whatever sticks out beyond the polygon as a negative triangle, and the area of the polygon itself is what remains.

Checking by cutting along diagonals

Triangle	Area
$\triangle ABC$	$6$
$\triangle ACD$	$\dfrac{19}{2}$
$\triangle ADE$	$\dfrac{9}{2}$
Total	$20$

The answers agree.

Cautions in use

The vertices must be listed in the order they are traced around the figure; reordering them gives a different shape
No side may cross another, since a self-intersecting outline makes the areas cancel
The polygon need not be convex

What the sign means

The sign of the value before the absolute value records the orientation: positive counterclockwise and negative clockwise. Computers also use that sign to decide which side of a polygon faces outward.

The five lines on the graph are the five sides, and the large dots are the five vertices.

Shoelace formula, Wikipedia