Revista Tocantinense de Matematica

Vol. 1, n. 1, 2026, p. 1–14

DOI: 10.20873/retmat.uft.v1n1.2026.23266

Received: 24 de junho de 2026

Accepted: 01 de julho de 2026

From Green’s Theorem to the Shoelace Formula: An

Application to the Calculation of the Area of the State of

Tocantins

Dimas Rosa Pimentel1*, Thiago Rodrigues Cavalcante2, Edcarlos Domingos da Silva 3

Abstract

This article develops a deduction of the Shoelace Formula from Green’s Theorem, highlight-

ing the relationships between determinants, line integrals, and the calculation of areas of

polygonal regions. First, the geometric interpretation of determinants is discussed as a tool

for area calculation, establishing connections between concepts from Analytic Geometry

and Vector Calculus. The Shoelace Formula is then deduced by parametrizing the edges of

a simple polygon and applying Green’s Theorem. As an application, the method is used to

estimate the area of the State of Tocantins by discretizing its boundary into points in the

Cartesian plane, yielding a value close to the oﬃcially reported area.

Keywords: Shoelace Formula; Green’s Theorem; Determinants; Area Calculation; Mathe-

matics Teaching.

Contents

1 Introduction 1

2 Area of a Triangle and a Geometric Interpretation of Determinants 3

2.1 Signed Area ...................................... 6

3 Shoelace Formula for Calculating the Area of a Polygon 7

3.0.1 Area of the State of Tocantins ....................... 11

4 Conclusion 13

1 Introduction

The calculation of areas is one of the most traditional topics in Mathematics and is present

at diﬀerent stages of school education. From the early years of Basic Education, students

come into contact with procedures for determining the areas of plane ﬁgures, initially through

formulas associated with elementary shapes and, later, through more general approaches

1PROFMAT, UFT- Campus Arraias,Tocantins, Brasil..

E-mail: dimasrosapimentel@seduc.to.gov.br. ORCID: .

*Autor correspondente.

2PROFMAT, UFT- Campus Arraias,Tocantins, Brasil.

E-mail: thiago.cavalcante@mail.uft.edu.br. ORCID: .

3Universidade Federal de Goias, IME, Goiânia-GO, Brazil.

E-mail: edcarlos@ufg.br. ORCID: .

ReTMAT 2

involving coordinates, geometric transformations, and algebraic concepts. In this context, the

study of areas fosters connections between diﬀerent ﬁelds of Mathematics, making it possible

to establish links between Geometry, Algebra, and Analysis.

Among the concepts closely related to area calculation, determinants stand out. Moreover

to their importance in solving linear systems and in various applications of Linear Algebra,

determinants admit a geometric interpretation associated with the measurement of areas and

volumes. In particular, the absolute value of the determinant of a second-order matrix can

be interpreted as the area of a parallelogram generated by two vectors in the plane, a result

widely discussed in Linear Algebra textbooks [1,5,7].

Although the geometric interpretation of determinants is widely discussed in the Linear

Algebra literature, its applications to the calculation of polygonal areas are not always explored

in depth in materials intended for Basic Education. In particular, methods that allow the

area of a polygonal region to be determined directly from the coordinates of its vertices are

often presented in an operational manner, without a detailed discussion of their mathematical

foundations.

This observation became especially evident from the analysis of an instructional sequence

made available by the Tocantins State Department of Education (SEDUC-TO), in which the

calculation of polygonal areas was carried out using a procedure based exclusively on the

coordinates of the vertices. This procedure, also described by Silva [

], provides a simple

and eﬃcient way to determine the areas of convex polygons. However, the mathematical

justiﬁcation of the method and its connections with broader mathematical concepts are not

discussed.

In this context, an interest arose in investigating the theoretical foundations of this procedure,

known in the literature as the Shoelace Formula. The study revealed a rich connection between

determinants, area calculation, and line integrals, showing how the formula can be obtained

through an application of Green’s Theorem. These relationships illustrate how concepts

traditionally addressed at diﬀerent educational levels can be articulated within a unifying

mathematical perspective.

This article originates from an educational product developed within the dissertation

Matrices, Linear Systems, and Determinants in Basic Education: Concepts and Applications,

produced in the context of the National Professional Master’s Program in Mathematics

(PROFMAT). Its objective is to present the Shoelace Formula through its relationship with

Green’s Theorem, highlighting the geometric interpretations involved and illustrating its

application through the estimation of the area of the State of Tocantins. In doing so, the work

seeks to provide mathematics teachers with material that promotes a deeper understanding of

the mathematical foundations underlying a method frequently used in educational contexts.

Beyond its mathematical interest, the approach oﬀers opportunities to discuss connections

among topics that are usually taught separately in school and undergraduate curricula. By

relating determinants, analytic geometry, line integrals, and area calculation, the proposed

discussion contributes to a more integrated view of Mathematics and highlights the potential

of interdisciplinary approaches in Mathematics Education.

The application developed in this work takes as its object of study the geographic boundary

of the State of Tocantins, making it possible to associate abstract mathematical concepts with a

context close to the reality of students and teachers in the region. In this sense, the investigation

contributes to bringing together concepts from Linear Algebra, Analytic Geometry, and Vector

Calculus through a concrete situation, favoring a contextualized approach aligned with the

appreciation of regional themes.

Although the Shoelace Formula is often presented as an algorithmic procedure for calculating

the area of simple polygons, its mathematical foundation is not always explored in teaching

ReTMAT 3

materials. Works such as those by Braden [

], Bourke [

], and Santos [

] highlight the

eﬃciency and elegance of the method, while Vector Calculus texts, such as Thomas, Weir, and

Hass [

], make it possible to understand the formula as a natural consequence of Green’s

Theorem. From this perspective, the present work seeks to bring these diﬀerent approaches

together, emphasizing the relationships among determinants, analytic geometry, line integrals,

and area calculation.

The distinguishing feature of this study lies in presenting an accessible exposition of the

Shoelace Formula based on concepts from Vector Calculus, preserving mathematical rigor while

emphasizing its didactic potential. Furthermore, the application of the method to estimate the

area of the State of Tocantins illustrates how abstract mathematical concepts can be used to

investigate problems related to the regional context, fostering an integrated approach between

theory and application.

The article is organized as follows. Section 2 presents the geometric interpretation of

determinants through the calculation of the area of triangles in the Cartesian plane, establishing

results that will serve as a basis for subsequent discussions. Section 3 develops the Shoelace

Formula through geometric arguments and subsequently presents its connection with Green’s

Theorem, culminating in an application to estimate the area of the State of Tocantins. Finally,

Section 4 presents the concluding remarks and some reﬂections on the pedagogical potential of

the approach developed.

Area of a Triangle and a Geometric Interpretation of Deter-

minants

As discussed in the introduction, determinants play an important role not only in Linear

Algebra but also in Geometry. One of their best-known geometric interpretations is related

to area calculation in the Cartesian plane. In particular, the determinant of a second-order

matrix can be interpreted as the signed area of the parallelogram generated by two vectors in

the plane, a result widely explored in Linear Algebra textbooks [1,5,7].

From this interpretation, it is possible to obtain formulas for calculating the areas of

polygonal ﬁgures using only the coordinates of their vertices. As a starting point for the

discussions developed in this work, we present a formula for the area of a triangle whose proof

highlights the close relationship between determinants and geometric concepts.

Proposition 2.1. Let

(

x1, y1

(

x2, y2

), and

(

x3, y3

)be the vertices of a triangle in the

Cartesian plane. Then its area is given by

Area =1

2



x1y11

x2y21

x3y31



Proof.

Without loss of generality, suppose that

x1≤x3≤x2

and that (

x3, y3

)lies above the

line segment connecting (x1, y1)and (x2, y2).

ReTMAT 4

(x1, y1)

(x2, y2)

(x3, y3)

(x1,0) (x2,0)(x3,0)

By projecting the vertices of the triangle onto the

-axis, we obtain three trapezoids whose

areas allow us to decompose the area of the triangle in terms of elementary ﬁgures.

.Trapezoid 1:(x1,0),(x1, y1),(x3, y3),(x3,0)

(x1, y1)

(x2, y2)

(x3, y3)

(x1,0) (x3,0) (x2,0)

.Trapezoid 2:(x3,0),(x3, y3),(x2, y2),(x2,0)

(x1, y1)

(x2, y2)

(x3, y3)

(x1,0) (x3,0) (x2,0)

.Trapezoid 3:(x1,0),(x1, y1),(x2, y2),(x2,0)

ReTMAT 5

(x1, y1)

(x2, y2)

(x3, y3)

(x1,0) (x3,0) (x2,0)

Using the classical formula for the area of a trapezoid, that is,

(B+b)h

where

and

represent the parallel bases and hthe height, we obtain

. Area of Trapezoid 1:Aτ1=1

2(y3+y1)(x3−x1)

. Area of Trapezoid 2:Aτ2=1

2(y3+y2)(x2−x3)

. Area of Trapezoid 3:Aτ3=1

2(y2+y1)(x2−x1)

Note that the union of trapezoids

τ1

and

τ2

contains exactly the desired triangular region plus

trapezoid

τ3

. Thus,

Aτ1

Aτ2

A∆

Aτ3,

from which it follows that

A∆

Aτ1

Aτ2−Aτ3.

In other words, we have

A∆=1

2[(y3+y1)(x3−x1)+(y3+y2)(x2−x3)−(y2+y1)(x2−x1)]

2[



y3x3−y3x1+y1x3−



y1x1+y3x2−



y3x3+



y2x2−y2x3−



y2x2+y2x1−y1x2+



y1x1]

2[x1y2+x2y3+x3y1−(x1y3+x2y1+x3y2)]

2(x1y2+x2y3+x3y1−x1y3−x2y1−x3y2)

2det 





x1y11

x2y21

x3y31





.

Since the area is always non-negative, we conclude that

A∆=1

2



det 





x1y11

x2y21

x3y31









The previous result shows that the area of a triangle can be calculated directly from the

coordinates of its vertices. More than a convenient formula, this result highlights the geometric

interpretation of determinants as signed area measures, a concept that will be fundamental for

the derivation of the Shoelace Formula presented in the following sections.

Note that the expression obtained in Proposition 2.1 can take positive or negative values,

depending on the order in which the triangle’s vertices are considered. When the vertices

ReTMAT 6

are traversed counterclockwise, the determinant is positive; when traversed clockwise, the

determinant is negative. Thus, the absolute value of the determinant gives the geometric

area of the triangle, while its sign indicates the orientation of the vertices. To illustrate this

geometric interpretation of determinants, we present below an example of calculating the area

of a triangle in the Cartesian plane from the coordinates of its vertices.

To consolidate the geometric interpretation of the determinant as an area measure, we

present below an example involving the calculation of the area of a triangle in the Cartesian

plane using determinants. This procedure highlights the relationship between the coordinates

of the vertices and the resulting value obtained for the area of the ﬁgure.

Example 2.2. We will determine the area of the triangular region whose vertices are the

points A(4,0),B(−1,1), and C(3,3). First, we represent the triangle in the Cartesian plane.

A(4,0)

B(−1,1)

C(3,3)

Applying the formula established in the previous proposition, we obtain

A∆=1

2



4 0 1

−111

3 3 1



2|4(1 ·1−3·1) + 3((−1) ·1−1·3) + 1((−1) ·3−1·3)|

2|−14|

= 7.

Therefore, the area of the triangular region is A∆= 7 a.u.

Note that the determinant involved in the previous example can take positive or negative

values, depending on the order in which the vertices are considered. Although the absolute

value is necessary to obtain the geometric area of the triangular region, the sign of the

determinant contains relevant information about the orientation of the path taken along the

vertices. This observation naturally leads to the concept of signed area, a fundamental tool for

the study of polygon areas and for the derivation of the Shoelace Formula.

2.1 Signed Area

The notion of signed area plays a central role in diﬀerent areas of Mathematics, especially

in Analytic Geometry, Linear Algebra, and Vector Calculus. As discussed by Anton [

], Larson

[

], and Hefez [

], the sign of the determinant is directly associated with the orientation of the

vectors that deﬁne it, making it possible to distinguish counterclockwise paths from clockwise

ones.

In the Cartesian plane, this interpretation establishes a natural convention: paths traversed

counterclockwise are associated with positive values of signed area, while paths traversed clock-

wise receive negative values. Thus, signed area simultaneously carries geometric information

ReTMAT 7

and information about the orientation of the boundary, an aspect that will be essential in

generalizing to the calculation of polygon areas [4,13].

Deﬁnition 2.3. Let

(

x1, y1

(

x2, y2

), and

(

x3, y3

)be three distinct points in the Cartesian

plane. The signed area of triangle ABC is deﬁned as the real number

Ao(ABC) = 1

2



x1y11

x2y21

x3y31



The geometric area of the triangle is obtained by

A(ABC) = |Ao(ABC)|.

The orientation of the triangle is said to be positive when the vertices are traversed

counterclockwise and negative when traversed clockwise.

Remark 2.4.The signed area does not depend on the geometric shape of the triangle, but on

the order of the vertices chosen for its algebraic representation. Thus, cyclic permutations of

the vertices preserve the value of

, while reversing the order of the vertices only changes the

sign of the expression.

In Example 2.2, for the triangle with vertices

0),

(

−

1), and

3), direct

application of the determinant expression shows that the value obtained for the area depends

on the order in which the vertices are taken. When traversed in the order

A→B→C

, a

positive quantity is obtained, while reversing this order produces the same value in absolute

terms, but with a negative sign. This observation conﬁrms, in a concrete way, that the

determinant is not limited to calculating the geometric area, but also incorporates algebraic

information associated with the orientation of the vertices in the Cartesian plane.

The deﬁnition of signed area allows composite regions to be treated algebraically, through

the sum of signed contributions. This additive structure will be used in the next section for

the case of simple polygons, naturally leading to the Shoelace Formula as a global expression

for calculating areas from vertices.

3 Shoelace Formula for Calculating the Area of a Polygon

As seen previously, the area of a triangle can be obtained from the coordinates of its vertices

through determinants, as presented in Larson [

]. The extension of this procedure to polygons

can be carried out by decomposing the region into triangles, so that the total area is obtained

by summing the areas of the constituent parts. Although conceptually simple, this method

becomes increasingly ineﬃcient as the number of vertices of the polygon increases, requiring a

large number of intermediate calculations.

In this context, the Shoelace Formula stands out as a direct alternative for calculating the

area of simple polygons from the ordered coordinates of their vertices. Motivated by Santos

[

], this formula makes it possible to reduce the problem to elementary algebraic operations

involving only the polygon’s vertices.

From a theoretical standpoint, this result can be interpreted as a consequence of Green’s

Theorem, as presented in Thomas [

], by transforming a line integral along the polygon’s

boundary into a discrete expression involving its coordinates. This interpretation highlights

the deep relationship between Analytic Geometry, determinants, and Integral Calculus.

As highlighted by Bourke [

], the systematic organization of the polygon’s vertices makes it

possible to eliminate intermediate calculation steps, making the method particularly eﬃcient

ReTMAT 8

for polygons with a large number of sides. Beyond its operational simplicity, the Shoelace

Formula reveals an interesting connection between diﬀerent areas of Mathematics, especially

Linear Algebra and Vector Calculus.

Deﬁnition 3.1. Let

be a simple polygon in the Cartesian plane with ordered vertices

P={(x1, y1),(x2, y2),...,(xn, yn)},where (xn+1, yn+1)=(x1, y1)to close the polygon.

The area of Pcan be calculated using the Shoelace Formula, given by

A(P) = 1

2



i=1

(xiyi+1 −xi+1yi)



The Shoelace Formula is a consequence of Green’s Theorem, stated as follows:

Theorem 3.2 (Green’s Theorem (ﬂux-divergence form)).Let

be a simple closed curve,

positively oriented, and let

be the region bounded by

. Let

be a vector ﬁeld

whose functions

and

have continuous partial derivatives in an open region containing

Then the outward ﬂux of

along

equals the double integral of the divergence of

over

that is,

ZZR∂M

∂x +∂N

∂y dx dy =IC

M dy −N dx.

The derivation of this method relates to line integrals and area integrals. A particular

choice of vector ﬁeld in Green’s Theorem allows us to obtain a direct expression for calculating

the area of plane regions. This choice leads to the proof of the following corollary:

Corollary 3.3. If

is a simple closed curve, positively oriented, bounding a region

, then

the area of Rcan be expressed as

Area(R) = 1

2IC

x dy −y dx.

Proof. Consider the vector ﬁeld F=Mi +N j, deﬁned by

M(x, y) = −1

2yand N(x, y) = 1

2x.

Applying Green’s Theorem in its classical form:

M dx +N dy =ZZR∂N

∂x −∂M

∂y dx dy,

we have ∂N

∂x =1

2and ∂M

∂y =−1

Hence,

∂N

∂x −∂M

∂y = 1.

Therefore, IC

M dx +N dy =ZZR

1dx dy.

Using the fact that

Area

(

) =

ZZR

dx dy

and substituting

and

into the line integral,

we obtain

Area(R) = IC−1

2ydx +1

2xdy

2IC

x dy −y dx.

ReTMAT 9

This yields the integral expression for calculating the area of a plane region bounded by a

closed curve C, as presented in Thomas [13].

Moving on to the derivation of the Shoelace Formula from Green’s Theorem, we will

transform the line integral into a sum over the polygon’s vertices. To do this, the polygon’s

boundary will be decomposed into line segments, each associated with a consecutive edge. By

parametrizing each edge of the polygon, that is, each line segment [(

xi, yi

)

(

xi+1, yi+1

)]

obtain a linear representation of the coordinates as a function of a parameter

t∈

1]. More

precisely, each edge of the polygon can be described by

γi

(

)=(

(

)

, y

(

)) = (

(

xi+1 −

)

, yi

(

yi+1 −yi

))

Now, diﬀerentiating

γi

(

)with respect to

and applying the chain rule,

we have

dx = (xi+1 −xi)dt and dy = (yi+1 −yi)dt.

Working now with the line integral, taking each γi(t)as the curve C, we obtain

Zγi

(x dy −y dx) = Z1

0h(xi+t(xi+1 −xi))(yi+1 −yi)−(yi+t(yi+1 −yi))(xi+1 −xi)idt.

Taking into account that this and noting the cancellation of the terms that depend on

, it

follows that Zγi

(x dy −y dx) = (xiyi+1 −xi+1yi).

Summing over all paths γi(t), that is, over the polygon’s edges, we obtain

(x dy −y dx) =

i=1

(xiyi+1 −xi+1yi).

Therefore, by Green’s Theorem,

A(P) = 1

2IC

(x dy −y dx) = 1

2



i=1

(xiyi+1 −xi+1yi)



The expression (

xiyi+1 −xi+1yi

)can be interpreted as the second-order determinant

associated with the position vectors of consecutive points of the polygon, that is,

xiyi+1 −xi+1yi=



xixi+1

yiyi+1



Thus, the area of the polygon can be written as the sum of the determinants associated

with its edges, that is,

A=1

i=1



xixi+1

yiyi+1



Expanding each determinant, we obtain

A=1

2[(x1y2+x2y3+···+xny1)−(y1x2+y2x3+· · · +ynx1)] .

On the other hand, this expression can be rearranged cyclically into a single determinant-like

structure involving the polygon’s vertices, resulting in the compact form

A=1

2



i=1

(xiyi+1 −xi+1yi)



ReTMAT 10

Thus, we obtain the Shoelace Formula for calculating the area of a simple polygon.

The name "shoelace formula" arises precisely from the pattern formed by the cross products

of the coordinates. By connecting the terms diagonally, as indicated by the arrows in the

scheme below, the products interlace in a manner similar to a shoelace being tied.

In practical terms, the procedure can be understood in three simple steps:

1. Arrange the coordinates of the vertices in two rows;

2. Repeat the ﬁrst point at the end;

Multiply diagonally, adding the products of the diagonals going from left to right (the

blue arrows) and subtracting the sum of the products of the diagonals going from right

to left (the arrows shown in red).

Area =1

x1x2x3· · · xnx1

y1y2y3· · · yny1

This structure shows that the Shoelace Formula results from a sum of second-order de-

terminants formed by consecutive vertices of the polygon. This formula makes it possible to

calculate areas directly from the coordinates of the points, avoiding geometric decompositions

and making the procedure more systematic.

In the example that follows, we present the application of the Shoelace Formula to calculate

the area of a simple polygon, illustrating the arrangement of the coordinates and the cross-

product pattern between consecutive vertices.

Example 3.4. Consider the polygon with vertices

A(1,1), B(3,0), C(5,1), D(6,2), E(2,4).

The polygon with the given coordinates is represented as follows

A(1,1)

B(3,0)

C(5,1)

D(6,2)

E(2,4)

First, we will calculate the area of this polygon by dividing it into three triangles, computing

their areas, and adding them together to obtain the total area of the polygon. Thus, we have

ReTMAT 11

Area =1

2



111

621

241



2



111

511

621



2



111

301

511



Area =1

2·14 + 1

2·4 + 1

2·4

Area = 7 + 2 + 2

Area = 11a.u

Now, applying the Shoelace Formula, we have

Area =1

135621

101241

Area =1

2|(0+3+10+24+2)−(3+0+6+4+4)|

Area =1

2(39 −17) = 1

2·22 = 11a.u

Note that, while the decomposition into triangles requires intermediate steps, the Shoelace

Formula provides the result directly from the coordinates of the vertices.

We now present an application of the Shoelace Formula to estimate the area of the State of

Tocantins, based on an approximate polygonal representation of its boundary in the Cartesian

plane.

As a ﬁnal application of the results developed throughout this work, we consider the use

of the Shoelace Formula to estimate the areas of plane regions with complex boundaries. In

particular, a didactic application of geographic interest consists of approximating the area of

federative units from polygonal representations of their boundaries.

From this perspective, the following subsection presents an application of the methodology

to the State of Tocantins, illustrating how discretizing its boundary into points in the Cartesian

plane allows an area estimate to be obtained through elementary algebraic operations.

3.0.1 Area of the State of Tocantins

In this subsection, we present an application of the Shoelace Formula to estimate the area

of the State of Tocantins, based on an approximate polygonal representation of its boundary

in the Cartesian plane.

The central idea is to represent the state’s boundary by means of an ordered set of points

in the plane, so that the region bounded by these points can be treated as a simple polygon.

This discretization makes it possible to replace a continuous geometric problem with a discrete

ReTMAT 12

algebraic problem, in which the area can be obtained directly through cross products of the

coordinates of the vertices.

Below, we present the points used in the modeling and the direct application of the Shoelace

Formula to calculate the approximate area.

Example 3.5. Consider the points

20)

, B

20)

, C

17)

, D

14)

, E

, F

, G

, H

, J

, K

, L

(11

,−

, M

,−

, N

,−

, O

,−

, P

,−

10)

, Q

(

−

,−

, R

(

−

,−

10)

(

−

,−

10)

, T

(

−

,−

, U

(

−

,−

10)

, V

(

−

,−

, W

, X

(

−

10)

, Y

14)

, Z

18)

which

deﬁne a polygonal approximation of the boundary of the State of Tocantins.

We wish to estimate its area using the Shoelace Formula.

A(0,20) B(3,20)

C(5,17)

D(4,14)

E(5,8) F(7,9)

G(8,8)

H(7,7)

I(7,5)

J(8,2)

K(8,0)

L(11,−2)

M(8,−3)

N(9,−5)

O(8,−8)

P(4,−10)

Q(−1,−9)

R(−4,−10)S(−6,−10)

T(−7,−9)

U(−8,−10)

V(−6,−2)

W(0,9)

X(−1,10)

Y(0,14)

Z(2,18)

For the calculation, the vertices are arranged in cyclic order, and the Shoelace Formula

is applied directly, summing the cross products of consecutive coordinates. Applying the

Shoelace Formula, we obtain

ReTMAT 13

Area =1

0 3 5 4 5 7 8 7 7 8 8 11 8 9 8 4 −1−4−6−7−8−60−10 2 0

20 20 17 14 8 9 8 7 5 2 0 −2−3−5−8−10−9−10−10−9−10−29 10 14 18 20

Area =1

2|244 −799|=1

2|−555|

2·555

= 277.5a.u

The value obtained provides a good approximation of the actual area of the State of Tocantins,

whose oﬃcial value is approximately 277

621

km2

. This result highlights the eﬃciency of

the Shoelace Formula as a tool for estimating areas of complex regions from a polygonal

discretization.

4 Conclusion

The Shoelace Formula is usually presented as an algorithmic procedure for calculating

the area of simple polygons, especially in contexts of Analytic Geometry and computational

applications. However, its mathematical foundation is not always explored in teaching materials,

which can limit conceptual understanding of the method.

In this work, we presented a derivation of the Shoelace Formula from Green’s Theorem,

showing that this expression is not merely an operational rule, but a direct consequence of

results from Vector Calculus. To this end, we ﬁrst explored the geometric interpretation of

determinants in the area of triangles and then the parametrization of the edges of a simple

polygon, which allowed the formula to be obtained systematically.

The application to the boundary of the State of Tocantins illustrated the eﬃciency of the

method in situations involving a large number of vertices, providing an estimate consistent

with the actual area of the region. This example reinforces the potential of the Shoelace

Formula as a mathematical modeling tool and as a didactic resource for integrating Linear

Algebra, Analytic Geometry, and Calculus.

It is hoped that this work will contribute to the education of teachers and students,

broadening the theoretical understanding behind procedures frequently used in the classroom.

As future perspectives, we suggest a deeper analysis of Green’s Theorem and its geometric

applications, as well as the exploration of extensions of the Shoelace Formula to more general

curves and area-calculation problems in more complex contexts.

ReTMAT 14

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