Shoelace formula

The shoelace formula, or shoelace algorithm, is a mathematical algorithm to determine the area of a polygon whose vertices are described by ordered pairs in the plane [http://staff.imsa.edu/math/journal/volume2/articles/Shoelace.pdf] . The user cross-multiplies corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon to find the area of the polygon within. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like tying shoelaces [ [http://www.mathreference.com/la-det,shoe.html Shoelace Formula ] ] . It is also sometimes called the shoelace method.The formula can be represented by the expression

: $mathbf\{A\}\; =\; \{1\; over\; 2\}|x\_1y\_2\; +\; x\_2y\_3\; +\; cdots\; +\; x\_\{n-1\}y\_n\; +\; x\_ny\_1\; -\; x\_2y\_1\; -\; x\_3y\_2\; -\; cdots\; -\; x\_ny\_\{n-1\}\; -\; x\_1y\_n|.$ [ Geometry for Enjoyment and Challenge section 16.2 ]

Examples

The user must know the points of the polygon in a Cartesian plane. For example, take a triangle with coordinates {(2, 1), (4, 5), (7, 8)}. Take the first "x"-coordinate and multiply it by the second "y"-value, and repeat, and keep repeating this process. This can be defined by this formula:

: $mathbf\{A\}\_\; ext\{tri.\}\; =\; \{1\; over\; 2\}|x\_1y\_2\; +\; x\_2y\_3\; +\; x\_3y\_1\; -\; x\_2y\_1\; -\; x\_3y\_2\; -\; x\_1y\_3|,$

for "x"_"n" and "y"_"n" representing each respective coordinate. Note that this works only for triangles. The number of sides of the polygon will alternate with the number of variables. For example, a pentagon (five sides) will be defined up to "x"₅ and "y"₅ points, up to

: $mathbf\{A\}\_\; ext\{pent.\}\; =\; \{1\; over\; 2\}|x\_1y\_2\; +\; x\_2y\_3\; +\; x\_3y\_4\; +\; x\_4y\_5\; +\; x\_5y\_1\; -\; x\_2y\_1\; -\; x\_3y\_2\; -\; x\_4y\_3\; -\; x\_5y\_4\; -\; x\_1y\_5|$

and a quadrilateral is

: $mathbf\{A\}\_\; ext\{quad.\}\; =\; \{1\; over\; 2\}|x\_1y\_2\; +\; x\_2y\_3\; +x\_3y\_4\; +\; x\_4y\_1\; -\; x\_2y\_1\; -\; x\_3y\_2\; -\; x\_4y\_3\; -\; x\_1y\_4|.$

One must take the absolute value of the answer before dividing by two.

Why "shoelace"

The reason this formula is called the shoelace formula is because of the method used to find it. This method uses matrices. If you were to find the area of a triangle with vertices (2,4), (3,−8), and (1,2), you would construct the following matrix by “walking around” your triangle and ending with the point you started with.

: [ IMSA JHMC Guide, Page. 10 "Shoelace" by Cindy Xi]

First, you draw diagonal down and to the right slashes, and multiply the two numbers connected by each slash, then add all the products: (2 × −8) + (3 × 2) + (1 × 4) = −6. Do the same with slashes diagonal down and to the left: (4 × 3) + (−8 × 1) + (2 × 2) = 8. Then, subtract these two numbers and take the absolute value of the difference: |−6 − 8| = 14. If you halve this, you have the area of the triangle: 7. If you consider which numbers you multiply and add, you will see how the formula relates to the matrix method. If you drew the slashes, you should see the shoelaces the formula is named for.

References