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
.