Shoelace formula

This MedLibrary.org supplementary page on Shoelace formula is provided directly from the open source Wikipedia as a service to our readers. Please see the note below on authorship of this content, as well as the Wikipedia usage guidelines. To search for other content from our encyclopedia supplement, please use the form below:

Related Sponsors

BM Pharmacy Generic pharmaceuticals at unbeatable prices. Insomnia, men's health, hair health, pain control. bmpharmacy.com
Online Pharmacy Carisoprodol, tadalafil, sildenafil, vardenafil and more... xlpharmacy.com
MedBasket.com Discreet. Inexpensive. Fast shipping. Great selection. What more can we say? medbasket.com
Frugalmed 2000 reasons to shop with us first. frugalmed.com
Ads by Tiva

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 plane1. 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 shoelaces2. 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| where n is the number of sides of the polygon.3

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}_\text{tri.} = {1 \over 2}|x_1y_2 + x_2y_3 + x_3y_1 - x_2y_1 - x_3y_2 - x_1y_3|

for xn and yn representing each respective coordinate. Note that this works only for triangles. Using this forumla, one finds that the area of the triangle equals one half of the absolute value of 10 + 32 + 7 - 4 - 35 - 16, which equals 3. 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 x5 and y5 points, up to

\mathbf{A}_\text{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}_\text{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.

\begin{bmatrix} 2 & 4 \\ 3 & -8 \\ 1 & 2 \\ 2 & 4 \end{bmatrix} 4

First, you draw diagonal down and to the right slashes (as shown below),

Image:ShoelaceMatrix2.GIF

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 (shown below with former slashes):

Image:ShoelaceMatrix3.GIF

(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 (above).

References

  1. ^ http://staff.imsa.edu/math/journal/volume2/articles/Shoelace.pdf
  2. ^ Shoelace Formula
  3. ^ Geometry for Enjoyment and Challenge section 16.2
  4. ^ IMSA JHMC Guide, Page. 10 "Shoelace" by Cindy Xi

Wikipedia content modification information:

  • This page was last modified on 3 November 2008, at 00:06.

Wikipedia Authorship and Review

Wikipedia content provided here is not reviewed directly by MedLibrary.org. Wikipedia content is authored by an open community of volunteers and is not produced by or in any way affiliated with MedLibrary.org.

Wikipedia Usage Guidelines

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article on "Shoelace formula".

The URL for this specific entry is:

All Wikipedia text is available under the terms of the GNU Free Documentation License. (See Copyrights for details). Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc.