Difference between revisions of "Shoelace Theorem"

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The '''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a [[polygon]] given the [[Cartesian coordinate system | coordinates]] of its [[vertex|vertices]].
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The '''Shoelace Theorem''' is a nifty formula for finding the [[area]] of a simple [[polygon]] given the [[Cartesian coordinate system|coordinates]] of its [[vertex|vertices]].
  
 
==Theorem==
 
==Theorem==
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 (<math>A</math>) of <math>P</math> is
+
Suppose the polygon <math>P</math> has vertices <math>(x_1, y_1)</math>, <math>(x_2, y_2)</math>, ... , <math>(x_n, y_n)</math>, listed in clockwise order.  Then the area (<math>A</math>) of <math>P</math> is
  
<cmath>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|</cmath>
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<cmath>A = \dfrac{1}{2} \left|(x_1y_2 + x_2y_3 + \cdots + x_ny_1) - (y_1x_2 + y_2x_3 + \cdots + y_nx_1) \right|</cmath>
  
You can also go counterclockwise order, as long as you find the absolute value of the answer.
+
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,
 
The Shoelace Theorem gets its name because if one lists the coordinates in a column,
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
(a_1 &, b_1) \\
+
(x_1 &, y_1) \\
(a_2 &, b_2) \\
+
(x_2 &, y_2) \\
 
& \vdots \\
 
& \vdots \\
(a_n &, b_n) \\
+
(x_n &, y_n) \\
(a_1 &, b_1) \\
+
(x_1 &, y_1) \\
 
\end{align*}</cmath>
 
\end{align*}</cmath>
 
and marks the pairs of coordinates to be multiplied,  
 
and marks the pairs of coordinates to be multiplied,  
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for(int i=1; i < 6; ++i)
 
for(int i=1; i < 6; ++i)
 
{
 
{
   label(i==3 ? "$\vdots$" : "$a$",(0,-i*.7));
+
   label(i==3 ? "$\vdots$" : "$x$",(0,-i*.7));
   label(i==3 ? "$\vdots$" : "$b$",(1.2,-i*.7));
+
   label(i==3 ? "$\vdots$" : "$y$",(1.2,-i*.7));
 
   label(subscripts[i-1],(0,-i*.7),SE,fontsize(9pt));  
 
   label(subscripts[i-1],(0,-i*.7),SE,fontsize(9pt));  
 
   label(subscripts[i-1],(1.2,-i*.7),SE,fontsize(9pt));  
 
   label(subscripts[i-1],(1.2,-i*.7),SE,fontsize(9pt));  
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pair c=(1.2,0);
 
pair c=(1.2,0);
 
label("$-$",shift(c)*(1.2,-2.1));
 
label("$-$",shift(c)*(1.2,-2.1));
label("$A=$",shift(-c)*(0,-2.1));
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label("$A=\frac12$",shift(-c)*(0,-2.1));
 +
draw(shift(-1/3*c)*((0,-.5)--(0,-3.9)));
 +
draw(shift(13/3*c)*((0,-.5)--(0,-3.9)));
  
 
for(int i=1; i < 6; ++i)
 
for(int i=1; i < 6; ++i)
 
{
 
{
   label(i==3 ? "$\vdots$" : "$a$",shift(3*c)*(0,-i*.7));
+
   label(i==3 ? "$\vdots$" : "$x$",shift(3*c)*(0,-i*.7));
   label(i==3 ? "$\vdots$" : "$b$",shift(3*c)*(1.2,-i*.7));
+
   label(i==3 ? "$\vdots$" : "$y$",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)*(0,-i*.7),SE,fontsize(9pt));  
 
   label(subscripts[i-1],shift(3*c)*(1.2,-i*.7),SE,fontsize(9pt));  
 
   label(subscripts[i-1],shift(3*c)*(1.2,-i*.7),SE,fontsize(9pt));  
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This can also be written in form of a summation <cmath>A = \dfrac{1}{2} \left|\sum_{i=1}^n{(x_{i+1}+x_i)(y_{i+1}-y_i)}\right|</cmath>
 
This can also be written in form of a summation <cmath>A = \dfrac{1}{2} \left|\sum_{i=1}^n{(x_{i+1}+x_i)(y_{i+1}-y_i)}\right|</cmath>
 
or in terms of determinants as <cmath>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|</cmath>
 
or in terms of determinants as <cmath>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|</cmath>
which is useful in the <math>3D</math> variant of the Shoelace theorem,
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which is useful in the <math>3D</math> variant of the Shoelace theorem. Note here that <math>x_{n+1} = x_1</math> and <math>y_{n+1} = y_1</math>.
 +
 
 +
The formula may also be considered a special case of Green's Theorem
 +
 
 +
<cmath>\tilde{A}=\int \int \left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dxdy=\oint(Ldx+Mdy)</cmath>
  
or as the special case of Green's Theorem
 
<center>
 
<math>\tilde{A}=\iint\left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dxdy=</math><font size="6">∳</font><math>(Ldx+Mdy)</math>
 
</center>
 
 
where <math>L=-y</math> and <math>M=0</math> so <math>\tilde{A}=A</math>.
 
where <math>L=-y</math> and <math>M=0</math> so <math>\tilde{A}=A</math>.
  
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Writing the coordinates in 3D and translating <math>\triangle ABC</math> so that <math>A=(0, 0, 0)</math> we get the new coordinates <math>A'(0, 0, 0)</math>, <math>B(x_2-x_1, y_2-y_1, 0)</math>, and <math>C(x_3-x_1, y_3-y_1, 0)</math>. Now if we let <math>\vec{b}=(x_2-x_1 \quad y_2-y_1 \quad 0)</math> and <math>\vec{c}=(x_3-x_1 \quad y_3-y_1 \quad 0)</math> then by definition of the cross product <math>[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}</math>.
 
Writing the coordinates in 3D and translating <math>\triangle ABC</math> so that <math>A=(0, 0, 0)</math> we get the new coordinates <math>A'(0, 0, 0)</math>, <math>B(x_2-x_1, y_2-y_1, 0)</math>, and <math>C(x_3-x_1, y_3-y_1, 0)</math>. Now if we let <math>\vec{b}=(x_2-x_1 \quad y_2-y_1 \quad 0)</math> and <math>\vec{c}=(x_3-x_1 \quad y_3-y_1 \quad 0)</math> then by definition of the cross product <math>[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}</math>.
 +
:)
  
 
===Proof:===
 
===Proof:===
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<cmath>=\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)}</cmath>
 
<cmath>=\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)}</cmath>
  
As claimed.
+
as claimed.
  
 
~ShreyJ
 
~ShreyJ
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\int_{\Omega}\alpha=\int_{\Omega}d\omega.
 
\int_{\Omega}\alpha=\int_{\Omega}d\omega.
 
</cmath>
 
</cmath>
Next, we use the theorem of Stokes to obtain
+
Next, we use the Theorem of Stokes to obtain
 
<cmath>
 
<cmath>
 
\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.
 
\int_{\Omega}d\omega=\int_{\partial\Omega}\omega.
Line 130: Line 133:
 
This is a very nice approach that directly helps in understanding the sum as terms which are areas of trapezoids.
 
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.)
+
The proof is in this book:
https://cses.fi/book/book.pdf
+
https://cses.fi/book/book.pdf#page=281
  
(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.)
+
(The only thing that needs to be slightly modified 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 ==
 
== Problems ==
 
=== Introductory ===
 
=== Introductory ===
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 of a triangle|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>.
 +
 
 
=== Exploratory ===
 
=== Exploratory ===
 
Observe that <cmath>\frac12\left|\det\begin{pmatrix}
 
Observe that <cmath>\frac12\left|\det\begin{pmatrix}

Latest revision as of 15:32, 22 February 2024

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

Theorem

Suppose the polygon $P$ has vertices $(x_1, y_1)$, $(x_2, y_2)$, ... , $(x_n, y_n)$, listed in clockwise order. Then the area ($A$) of $P$ is

\[A = \dfrac{1}{2} \left|(x_1y_2 + x_2y_3 + \cdots + x_ny_1) - (y_1x_2 + y_2x_3 + \cdots + y_nx_1) \right|\]

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*} (x_1 &, y_1) \\ (x_2 &, y_2) \\ & \vdots \\ (x_n &, y_n) \\ (x_1 &, y_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 ? "$\vdots$" : "$x$",(0,-i*.7));   label(i==3 ? "$\vdots$" : "$y$",(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("$A=\frac12$",shift(-c)*(0,-2.1)); draw(shift(-1/3*c)*((0,-.5)--(0,-3.9))); draw(shift(13/3*c)*((0,-.5)--(0,-3.9)));  for(int i=1; i < 6; ++i) {   label(i==3 ? "$\vdots$" : "$x$",shift(3*c)*(0,-i*.7));   label(i==3 ? "$\vdots$" : "$y$",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.

Other Forms

This can also be written in form of a summation \[A = \dfrac{1}{2} \left|\sum_{i=1}^n{(x_{i+1}+x_i)(y_{i+1}-y_i)}\right|\] or in terms of determinants as \[A = \dfrac{1}{2} \left|\sum_{i=1}^n{\det\begin{pmatrix}x_i&x_{i+1}\\y_i&y_{i+1}\end{pmatrix}}\right|\] which is useful in the $3D$ variant of the Shoelace theorem. Note here that $x_{n+1} = x_1$ and $y_{n+1} = y_1$.

The formula may also be considered a special case of Green's Theorem

\[\tilde{A}=\int \int \left(\frac{\partial M}{\partial x}-\frac{\partial L}{\partial y}\right)dxdy=\oint(Ldx+Mdy)\]

where $L=-y$ and $M=0$ so $\tilde{A}=A$.

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.

The proof is in this book: https://cses.fi/book/book.pdf#page=281

(The only thing that needs to be slightly modified 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 $\triangle ABF$.

Exploratory

Observe that \[\frac12\left|\det\begin{pmatrix} x_1 & y_1\\ x_2 & y_2 \end{pmatrix}\right|\] is the area of a triangle with vertices $(x_1,y_1),(x_2,y_2),(0,0)$ and \[\frac16\left|\det\begin{pmatrix} x_1 & y_1 & z_1\\ x_2 & y_2 & z_2\\ x_3 & y_3 & z_3 \end{pmatrix}\right|\] is the volume of a tetrahedron with vertices $(x_1, y_1, z_1), (x_2, y_2, z_2),(x_3, y_3, z_3),(0,0,0)$. Does a similar formula hold for $n$Dimensional triangles for any $n$? If so how can we use this to derive the $n$D Shoelace Formula?

External Links

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

Nice geometric approach and discussion for proving the 3D Shoelace Theorem by Nicholas Patrick and Nadya Pramita: [2]

Nice integral approach for proving the 3D Shoelace Theorem (ignoring sign of volume) by @george2079: [3] AOPS