Difference between revisions of "Shoelace Theorem"
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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 the area of <math>P</math> 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 the area of <math>P</math> is | ||
− | <cmath>\dfrac{1}{2} |(a_1b_2 + a_2b_3 + \cdots + | + | <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> |
The Shoelace Theorem gets its name because if one lists the coordinates in a column, | The Shoelace Theorem gets its name because if one lists the coordinates in a column, |
Revision as of 09:31, 26 August 2012
The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.
Contents
[hide]Theorem
Suppose the polygon has vertices , , ... , , listed in clockwise order. Then the area of is
The Shoelace Theorem gets its name because if one lists the coordinates in a column, and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.
Proof
Let be the set of points belonging to the polygon. We have that where . The volume form is an exact form since , where Using this substitution, we have Next, we use the theorem of Green to obtain We can write , where is the line segment from to . With this notation, we may write If we substitute for , we obtain If we parameterize, we get Performing the integration, we get More algebra yields the result
Problems
Introductory
In right triangle , we have , , and . Medians and are drawn to sides and , respectively. and intersect at point . Find the area of .