Difference between revisions of "Shoelace Theorem"

m
Line 1: Line 1:
'''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a [[polygon]] given the coordinates of it's [[vertex|vertices]].
+
The '''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a [[polygon]] given the [[Cartesian coordinate system | coordinates]] of it's [[vertex|vertices]].
  
 
==Theorem==
 
==Theorem==
Let the coordinates, in "clockwise" order, be <math>(a_1, b_1)</math>, <math>(a_2, b_2)</math>, ... , <math>(a_n, b_n)</math>. The area of the polygon is
+
Suppose the polygon <math>P</math> has vertices <math>(a_1, b_1)</math>, <math>(a_2, b_2)</math>, ... , <math>(a_n, b_n)</math>, listed in clockwise order. Then area of <math>P</math> is
  
 
<cmath>\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2-b_2a_3-\cdots -b_na_1|.</cmath>
 
<cmath>\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2-b_2a_3-\cdots -b_na_1|.</cmath>
  
Shoelace Theorem gets it's name by listing the coordinates like so:
+
The Shoelace Theorem gets its name because if one lists the the coordinates in a column,
 
+
<cmath>\begin{align*}
<cmath>(a_1, b_1)</cmath>
+
(a_1 &, b_1) \\
 
+
(a_2 &, b_2) \\
<cmath>(a_2, b_2)</cmath>
+
& \vdots \\
 
+
(a_n &, b_n) \\
<cmath>\vdots</cmath>
+
(a_1 &, b_1)
 
+
\end{align*},</cmath>
<cmath>(a_n, b_n)</cmath>
+
and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
 
 
<cmath>(a_1, b_1)</cmath>
 
  
 
==Proof==
 
==Proof==

Revision as of 12:18, 24 April 2008

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

Theorem

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

Proof

Template:Incomplete

This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS