Difference between revisions of "Shoelace Theorem"
(→Proof 2) |
|||
Line 16: | Line 16: | ||
and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes. | and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes. | ||
+ | ==Proof 1== | ||
+ | Typing... ~ ShreyJ | ||
==Proof 2== | ==Proof 2== | ||
Let <math>\Omega</math> be the set of points belonging to the polygon. | Let <math>\Omega</math> be the set of points belonging to the polygon. |
Revision as of 13:18, 6 July 2018
The Shoelace Theorem is a nifty formula for finding the area of a polygon given the coordinates of its vertices.
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 1
Typing... ~ ShreyJ
Proof 2
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 Stokes 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
.
External Links
A good explanation and exploration into why the theorem works by James Tanton: [1] AOPS