Difference between revisions of "Shoelace Theorem"

Revision as of 03:08, 24 December 2020

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

Theorem

Suppose the polygon $P$ has vertices $(a_1, b_1)$ , $(a_2, b_2)$ , ... , $(a_n, b_n)$ , listed in clockwise order. Then the area ( $A$ ) of $P$ is

$A = \dfrac{1}{2} \left|(a_1b_2 + a_2b_3 + \cdots + a_nb_1) - (b_1a_2 + b_2a_3 + \cdots + b_na_1) \right|$

You can also go counterclockwise order, as long as you find the absolute value of the answer.

The Shoelace Theorem gets its name because if one lists the coordinates in a column, $\begin{align*} (a_1 &, b_1) \\ (a_2 &, b_2) \\ & \vdots \\ (a_n &, b_n) \\ (a_1 &, b_1) \\ \end{align*}$ and marks the pairs of coordinates to be multiplied, $[asy] unitsize(1cm); string[] subscripts={"$1$","$2$"," ","$n$","$1$"}; for(int i=1; i < 6; ++i) { label(i==3 ? "$\vdots$" : "$a$",(0,-i*.7)); label(i==3 ? "$\vdots$" : "$b$",(1.2,-i*.7)); label(subscripts[i-1],(0,-i*.7),SE,fontsize(9pt)); label(subscripts[i-1],(1.2,-i*.7),SE,fontsize(9pt)); } for(int i=1; i<5; ++i) draw((0.3,-i*.7)--(1,-(i+1)*.7)); pair c=(1.2,0); label("$-$",shift(c)*(1.2,-2.1)); label("$A=$",shift(-c)*(0,-2.1)); for(int i=1; i < 6; ++i) { label(i==3 ? "$\vdots$" : "$a$",shift(3*c)*(0,-i*.7)); label(i==3 ? "$\vdots$" : "$b$",shift(3*c)*(1.2,-i*.7)); label(subscripts[i-1],shift(3*c)*(0,-i*.7),SE,fontsize(9pt)); label(subscripts[i-1],shift(3*c)*(1.2,-i*.7),SE,fontsize(9pt)); } for(int i=1; i<5; ++i) draw(shift(3*c)*(0.3,-(i+1)*.7)--shift(3*c)*(1,-i*.7)); [/asy]$ the resulting image looks like laced-up shoes.

This can also be written in form of a summation $A = \dfrac{1}{2} \left|\sum_{i=1}^n{(x_{i+1}+x_i)(y_{i+1}-y_i)}\right|$ And thus we can introduce determinants to get $A = \dfrac{1}{2} \left|\sum_{i=1}^n{\det\begin{pmatrix}x_i&x_{i+1}\\y_i&y_{i+1}\end{pmatrix}}\right|$ This is more helpful in the $3D$ formula variant of the Shoelace theorem.

Proof 1

Claim 1: The area of a triangle with coordinates $A(x_1, y_1)$ , $B(x_2, y_2)$ , and $C(x_3, y_3)$ is $\frac{|x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2|}{2}$ .

Proof of claim 1:

Writing the coordinates in 3D and translating $\triangle ABC$ so that $A=(0, 0, 0)$ we get the new coordinates $A'(0, 0, 0)$ , $B(x_2-x_1, y_2-y_1, 0)$ , and $C(x_3-x_1, y_3-y_1, 0)$ . Now if we let $\vec{b}=(x_2-x_1 \quad y_2-y_1 \quad 0)$ and $\vec{c}=(x_3-x_1 \quad y_3-y_1 \quad 0)$ then by definition of the cross product $[ABC]=\frac{||\vec{b} \times \vec{c}||}{2}=\frac{1}{2}||(0 \quad 0 \quad x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)||=\frac{x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2}{2}$ .

Proof:

We will proceed with induction.

By claim 1, the shoelace theorem holds for any triangle. We will show that if it is true for some polygon $A_1A_2A_3...A_n$ then it is also true for $A_1A_2A_3...A_nA_{n+1}$ .

We cut $A_1A_2A_3...A_nA_{n+1}$ into two polygons, $A_1A_2A_3...A_n$ and $A_1A_nA_{n+1}$ . Let the coordinates of point $A_i$ be $(x_i, y_i)$ . Then, applying the shoelace theorem on $A_1A_2A_3...A_n$ and $A_1A_nA_{n+1}$ we get

$[A_1A_2A_3...A_n]=\frac{1}{2}\sum_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)$ $[A_1A_nA_{n+1}]=\frac{1}{2}(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)$

Hence

$[A_1A_2A_3...A_nA_{n+1}]=[A_1A_2A_3...A_n]+[A_1A_nA_{n+1}]=\frac{1}{2}\sum_{i=1}^{n}(x_iy_{i+1}-x_{i+1}y_i)+\frac{1}{2}(x_1y_2+x_2y_3+x_3y_1-x_1y_3-x_2y_1-x_3y_2)$ $=\frac{1}{2}((x_2y_1+x_3y_2+...+x_{n+1}y_n+x_1y_{n+1})-(x_1y_2+x_2y_3+...+x_ny_{n+1}+x_{n+1}y_1))=\boxed{\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i)}$

As claimed.

~ShreyJ

Proof 2

Let $\Omega$ be the set of points belonging to the polygon. We have that $A=\int_{\Omega}\alpha,$ where $\alpha=dx\wedge dy$ . The volume form $\alpha$ is an exact form since $d\omega=\alpha$ , where $\omega=\frac{x\,dy}{2}-\frac{y\,dx}{2}.\label{omega}$ Using this substitution, we have $\int_{\Omega}\alpha=\int_{\Omega}d\omega.$ Next, we use the theorem of Stokes to obtain $\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.$ We can write $\partial \Omega=\bigcup A(i)$ , where $A(i)$ is the line segment from $(x_i,y_i)$ to $(x_{i+1},y_{i+1})$ . With this notation, we may write $\int_{\partial\Omega}\omega=\sum_{i=1}^n\int_{A(i)}\omega.$ If we substitute for $\omega$ , we obtain $\sum_{i=1}^n\int_{A(i)}\omega=\frac{1}{2}\sum_{i=1}^n\int_{A(i)}{x\,dy}-{y\,dx}.$ If we parameterize, we get $\frac{1}{2}\sum_{i=1}^n\int_0^1{(x_i+(x_{i+1}-x_i)t)(y_{i+1}-y_i)}-{(y_i+(y_{i+1}-y_i)t)(x_{i+1}-x_i)\,dt}.$ Performing the integration, we get $\frac{1}{2}\sum_{i=1}^n\frac{1}{2}[(x_i+x_{i+1})(y_{i+1}-y_i)- (y_{i}+y_{i+1})(x_{i+1}-x_i)].$ More algebra yields the result $\frac{1}{2}\sum_{i=1}^n(x_iy_{i+1}-x_{i+1}y_i).$

Proof 3

This is a very nice approach that directly helps in understanding the sum as terms which are areas of trapezoids.

See page 281 in this book (in the Polygon Area section.) https://cses.fi/book/book.pdf

(The only thing that needs to be modified in this proof is that one must shift the entire polygon up by k, until all the y coordinates are positive, but this term gets canceled in the resulting sum.)

Problems

Introductory

In right triangle $ABC$ , we have $\angle ACB=90^{\circ}$ , $AC=2$ , and $BC=3$ . Medians $AD$ and $BE$ are drawn to sides $BC$ and $AC$ , respectively. $AD$ and $BE$ intersect at point $F$ . Find the area of $\triangle ABF$ .

External Links

A good explanation and exploration into why the theorem works by James Tanton: [1] AOPS

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