Difference between revisions of "Shoelace Theorem"
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| − | '''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== | ||
| − | + | 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 | + | The Shoelace Theorem gets its name because if one lists the the coordinates in a column, |
| − | + | <cmath>\begin{align*} | |
| − | <cmath>(a_1, b_1) | + | (a_1 &, b_1) \\ |
| − | + | (a_2 &, b_2) \\ | |
| − | + | & \vdots \\ | |
| − | + | (a_n &, b_n) \\ | |
| − | + | (a_1 &, b_1) | |
| − | + | \end{align*},</cmath> | |
| − | + | and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes. | |
| − | |||
| − | |||
==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 
has vertices 
, 
, ... , 
, listed in clockwise order. Then area of is
![\[\dfrac{1}{2} |a_1b_2+a_2b_3+\cdots +a_nb_1-b_1a_2-b_2a_3-\cdots -b_na_1|.\]](http://latex.artofproblemsolving.com/0/0/f/00fb3ab55fb21e342cd434b951e5acea267f35cc.png)
The Shoelace Theorem gets its name because if one lists the the coordinates in a column,
and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
Proof
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