# Difference between revisions of "Shoelace Theorem"

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In right triangle <math>ABC</math>, we have <math>\angle ACB=90^{\circ}</math>, <math>AC=2</math>, and <math>BC=3</math>. [[Median]]s <math>AD</math> and <math>BE</math> are drawn to sides <math>BC</math> and <math>AC</math>, respectively. <math>AD</math> and <math>BE</math> intersect at point <math>F</math>. Find the area of <math>\triangle ABF</math>. | In right triangle <math>ABC</math>, we have <math>\angle ACB=90^{\circ}</math>, <math>AC=2</math>, and <math>BC=3</math>. [[Median]]s <math>AD</math> and <math>BE</math> are drawn to sides <math>BC</math> and <math>AC</math>, respectively. <math>AD</math> and <math>BE</math> intersect at point <math>F</math>. Find the area of <math>\triangle ABF</math>. | ||

+ | |||

+ | == External Links== | ||

+ | A good explanation and exploration into why the theorem works: | ||

+ | [http://www.jamestanton.com/wp-content/uploads/2012/03/Cool-Math-Essay_June-2014_SHOELACE-FORMULA.pdf] | ||

## Revision as of 13:34, 15 August 2015

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

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 .

## External Links

A good explanation and exploration into why the theorem works: [1]