Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Shoelace Theorem

Revision as of 12:05, 24 April 2008 by 1=2 (talk | contribs) (New page: '''Shoelace Theorem''' is a nifty formula for finding the area of a polygon given the coordinates of it's vertices. ==Theorem== Let the coordinates, in "clockwise" orde...)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of it's vertices.

Theorem

Let the coordinates, in "clockwise" order, be $(a_1, b_1)$, $(a_2, b_2)$, ... , $(a_n, b_n)$. The area of the polygon is

\[\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2-b_2a_3-\cdots -b_na_1|.\]

Shoelace Theorem gets it's name by listing the coordinates like so:

\[(a_1, b_1)\] \[(a_2, b_2)\] \[\vdots\] \[(a_n, b_n)\] \[(a_1, b_1)\]

Proof

Template:Incomplete

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Shoelace_Theorem&oldid=25246"