# Difference between revisions of "Shoelace Theorem"

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==Proof== | ==Proof== | ||

− | {{ | + | Let <math>\Omega</math> be the set of points belonging to the polygon. |

+ | We have that | ||

+ | <cmath> | ||

+ | A=\int_{\Omega}\alpha, | ||

+ | </cmath> | ||

+ | where <math>\alpha=dx\wedge dy</math>. | ||

+ | The volume form <math>\alpha</math> is an exact form since <math>d\omega=\alpha</math>, where | ||

+ | <cmath> | ||

+ | \omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega} | ||

+ | </cmath> | ||

+ | Using this substitution, we have | ||

+ | <cmath> | ||

+ | \int_{\Omega}\alpha=\int_{\Omega}d\omega. | ||

+ | </cmath> | ||

+ | Next, we use the theorem of Green to obtain | ||

+ | <cmath> | ||

+ | \int_{\Omega}d\omega=\int_{\partial\Omega}\omega. | ||

+ | </cmath> | ||

+ | We can write <math>\partial \Omega=\bigcup A(i)</math>, where <math>A(i)</math> is the line | ||

+ | segment from <math>(x_i,y_i)</math> to <math>(x_{i+1},y_{i+1})</math>. With this notation, | ||

+ | we may write | ||

+ | <cmath> | ||

+ | \int_{\partial\Omega}\omega=\sum_{i=1}^n\int_{A(i)}\omega. | ||

+ | </cmath> | ||

+ | If we substitute for <math>\omega</math>, we obtain | ||

+ | <cmath> | ||

+ | \sum_{i=1}^n\int_{A(i)}\omega=\frac{1}{2}\sum_{i=1}^n\int_{A(i)}{x\,dy}-{y\,dx}. | ||

+ | </cmath> | ||

+ | If we parameterize, we get | ||

+ | <cmath> | ||

+ | \frac{1}{2}\sum_{i=1}^n\int_0^1{(x_i+(x_{i+1}-x_i)t)(y_{i+1}-y_i)}-{(y_i+(y_{i+1}-y_i)t)(x_{i+1}-x_i)\,dt}. | ||

+ | </cmath> | ||

+ | Performing the integration, we get | ||

+ | <cmath> | ||

+ | \frac{1}{2}\sum_{i=1}^n\frac{1}{2}[(x_i+x_{i+1})(y_{i+1}-y_i)- | ||

+ | (y_{i}+y_{i+1})(x_{i+1}-x_i)]. | ||

+ | </cmath> | ||

+ | More algebra yields the result | ||

+ | <cmath> | ||

+ | \frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i). | ||

+ | </cmath> | ||

== Problems == | == Problems == |

## Revision as of 18:39, 28 February 2010

The **Shoelace Theorem** is a nifty formula for finding the area of a polygon given the coordinates of its vertices.

## Contents

## 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 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 .

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