This work is licensed under a Creative Commons

Attribution-NonCommercial-ShareAlike 4.0 International License.

F. Zastrow, L. Do Nascimento,

“Pappus-Guldin theorems applied to the study of solid modeling with GeoGebra software”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024

Pappus-Guldin theorems

applied to the study

of solid modeling with

GeoGebra software

ARTICLE HISTORY

Received 14 March 2023

Accepted 12 May 2023

Published 8 January 2024

Fernanda Zastrow Tavares

Universidade Federal do Rio Grande (FURG)

Rio Grande, Brasil

fernandazastrow@gmail.com

ORCID: 0000-0001-8323-4221

Luverci do Nascimento Ferreira

Universidade Federal do Rio Grande (FURG)

Rio Grande, Brasil

luverci@gmail.com

ORCID: 0000-0002-0662-9112

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

This work is licensed under a Creative Commons

Attribution-NonCommercial-ShareAlike 4.0 International License.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024

10.5281/zenodo. 10402067

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January - June 2024

Pappus-Guldin theorems applied the study of solid

modeling with GeoGebra software

Fernanda Zastrow Tavares

Universidade Federal do Rio Grande

(FURG)

Instituto de Matemática, Estatística e Fi-

sica (IMEF)

Rio Grande, Brasil

fernandazastrow@gmail.com

Luverci do Nascimento Ferreira

Universidade Federal do Rio Grande

(FURG)

Instituto de Matemática, Estatística e Fí-

sica (IMEF)

Rio Grande, Brasil

luverci@gmail.com

Abstract— In this work, we use Geogebra software to simulate

the shape of objects (solids) in three dimensions from their photo

and real dimensions using spline interpolation. With the recon-

structed object, we analyze its volume and surface area using the

Pappus-Guldin Theorems (PGT), the theorems that use mathemati-

cal analysis ideas to describe the volume and surface area by the sec-

tional area and by the contour curve of the object. In the simulations,

we tested the verification of the modeling for known solids (sphere

and torus) and then analyzed some objects used in the industry, such

as the packaging of products, pet bottles, yogurt containers, coffee

powder packaging, aluminum soda cans, and the packaging of choc-

olate powder. We also analyzed some objects created by rotating

bodies, such as the shape of a jar and an aluminum barrel, and also

shapes found in nature, such as the shape of a pear and an egg. Mod-

eling allows us to better understand the packaging used in the indus-

try to minimize manufacturing costs and maximize its utility. Thus,

we can modify these packages to obtain the best development of how

these products are presented to the public, optimizing its format by

analyzing its surface and its volume.

Keywords—Numerical simulation, GeoGebra, Spline, Pappus-

Guldin theorems.

I. I

NTRODUCTION

Applications involving solid geometric objects that relate

to the measurement of dimension or volume of a three-dimen-

sional shape are classic examples that, according to [12], re-

quire the study of geometry of these bodies. The simple ap-

plication of this work would be to calculate the area and vol-

ume of any three-dimensional object. This theoretical tool

that gives us the volume and surface area of bodies, with the

idea centered on axis rotation, are the Pappus-Guldin Theo-

rems (PGT). Automatic fulfillment of the requirements for

the use of electronic computers. This tool helps to understand

how computational modeling is performed in certain solid

models.

Based on the article [3] on rotational shapes and volume cal-

culation, we use the numerical simulation code. The applica-

tion will be a calculation of area and volume by computa-

tional modeling, starting from the analysis of an object from

a photograph, which is inserted into the GeoGebra program

according to [6] and using the spline command, can be iden-

tified the generation of contours of the object, building points

on the contours of the image is called a polygonal line. There-

fore, we will use the free software GeoGebra to model some

objects and find the volume and surface area of these objects,

the axis can be formed by rotation around a surface or curve.

Throughout the process, we have the minimum surfaces, and

surfaces of revolution have a limit that will be discussed so

that we can use the Pappus-Guldin theorems (PGT).

We will study these applications from the calculation of

surface area and volume. For example, two methods can be

compared: organic and industrial composting. In biological

and engineering applications, we can use the PGT model to

calculate the lateral surface area, and we can know the mass

in terms of light and amount of water it produces on the sur-

face in terms of the best fruit, flavor, and size. The industrial

application of packaging can help to recycle and is less harm-

ful to environment. We can minimize the product from the

packaging in formats: for less heat exchange at room temper-

ature; lower material costs. In the last part of the work, theo-

retical studies were carried out on the smallest surfaces of the

packaging to minimize the design process of the designers of

the formats of these packages.

II.

ETHODOLOGY

Numerical simulations of geometric solids using the mathe-

matical theory of PGT will be applied. This can result in a

study of minimizing the manufacturing costs of the products

of companies through the packaging format of their products.

The software used for the calculations, Geogebra, is free

software with mathematical functions developed by mathe-

matician Markus Hohenwarter as part of his Ph.D. thesis [6]

at the University of Salzburg. The program aims to develop

suitable tools for teaching mathematics and applying mathe-

matics in the fields of geometry and algebra.

In class, we use mathematical and physical concepts to

follow theory and practice. However, nowadays they are not

always together. There are often people who have the theory

and do not know how to put it into practice or vice versa.

Therefore, it shows the reality applied in daily practice

In this article, based on a new technology called Infor-

mation and Communication Technology (ICT), we use this

software to perform numerical simulations. GeoGebra in

which we will model command objects with splines accord-

ing to [4].

For theoretical support, we used the PGT to calculate the

surface area and volume of solids. These theorems were

proved by the Greek mathematician Pappus of Alexandria

and the Swiss mathematician Paul Goulding, who used the

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024

10.5281/zenodo. 10402067

F. Zastrow, L. Do Nascimento,

“Pappus-Guldin theorems applied to the study of solid modeling with GeoGebra software”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024

concept of the center of rotation of solids to perform analyses

and their measurements (see [9], [12], and [13]).

A. Pappus-Guldin theorems

There are two theorems referring to Pappus and Guldin:

First Pappus-Guldin theorem: Considering R as a plane

figure, the solid  formed by rotation of R around an axis r

has the volume of the solid is equal to area of R multiplied

by the length of circle described by its center of gravity (see

Fig.1).

The volume of the solid is thus given by the equation:

Vol(  ) = 2  dist(r, G) A(R), (3)

where G is the center of gravity of R, 2dist(r, G) is the dis-

tance of the line r from the center of gravity, and A(R) is the

area of the region R.

Fig.1. First Pappus-Guldin theorem

Second Pappus-Guldin theorem: Considering C as a plane

curve, the surface  formed by rotation of C around an axis

r has area obtained by multiplying the length of the curve C

by length of the circumference passing through its center of

gravity G (see Fig.2).

The surface area of  is given by the equation:

A()= 2  dist(r, G) l(C), (4)

where G is the center of gravity of curve C, 2  dist(r, G) is

the distance of the center of gravity of the lines r, and l(C) is

the length of curve C.

B. Spline

A spline is defined as a partitioned domain of a polyno-

mial of degree n whose function value and its n-1 continuous

first derivatives pass through the connection points. The ab-

scissa of these connecting points are called knots, and these

piecewise polynomials are chosen to minimize the least mean

square curvature.

According to [1], splines can be divided into two catego-

ries:

 Interpolation splines, which pass through all control

points.

 Approximation splines, which run near all control

points

Let a=



<



<...<



=b, be a subdivision of the interval

(a,b). A spline function of degree n with knots at the points



i=0,1,..., m is a function S with the following properties ac-

cording to [1]:

 In each subinterval (



, 



), i=0,1,..., m-1, S(x) is a

polynomial of degree n.

 S(x) and its first derivatives (n-1) are continuous on

the interval (a,b).

According to [8], we can use spline functions to calculate

tree volumes. The method used to calculate the volume of a

tree would be to calculate the volume from the diameter and

height of the trunk, using a cubic spline.

Fig.2. Second Pappus-Guldin theorem

III. N

UMERICAL

IMULATION

In this section, we will perform simulations using Geoge-

bra software and the Pappus-Guldin theorem (PGT) theory in

the study of the volume and surface area of sphere and torus

to verify the accuracy of the method.

A. Sphere

The 3D modeling of the sphere (see Fig.3) was performed

in GeoGebra software according to [6]. We note that as the

value of n increases, there is a built-in convergence for the

sphere with radius r=4. Given a value and knowing that r is

the fixed radius, we conclude that the volume of a sphere us-

ing the PGT (



) in the limit, approximates the volume

value of the sphere using known results, i.e.,









= 









(1)

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where the locus of r is constant according to [5]. These values

of absolute error and relative error decrease when only the

value of n increases. We applied the same procedure by cal-

culating a rough estimate of what was done for volume of the

sphere with a different algorithm for the area of sphere using

GeoGebra software from the creator of the application [6],

which confirms the same idea.

Fig. 3. Sphere simulation

Table I shows the algorithm of the PGT, which calculated

the volume of the sphere in GeoGebra software. In the numer-

ical simulation, we have a fixed radius of r=4, where we ob-

serve that the number of sides of the polygon inscribed in the

circle depends on the value of n. The values of n are chosen

randomly. Table I analyzes the value of the volume of the

sphere, which is estimated to be the 

ℎ

= 268.08 u.v.

Then we compare the value obtained over PGT volume with

the modeling, which converges when it is verified that the

value of n increases. We have below Table I:

TABLE

PHERE SIMULATION

Table 1: Sphere simulation

n VPGT Absolute error Relative error

54 267.86 0.22 0.08206505521

77 267.97 0.11 0.0410325276

91 268 0.08 0.02984183826

111 268.03 0.05 0.01865114891

B. Torus

The 3D modeling of the torus (see Fig.4) was performed

in GeoGebra software. The Cartesian plane coordinates x, y,

and z in three dimensions represented by 3D is in the interval

[0, 2], the torus has its rotational symmetry in the z-axis, R

is the distance from the center of the tube to the center of the

torus, and r is the radius of the tube.

The volume of the torus is given by:





=2



(



)(

2

)

(2)

Fig. 4. Torus simulation

Table II shows the PGT algorithm performed in GeoGe-

bra software for volume of a torus. In the numerical simula-

tion, we have fixed radius of r and s, while the values of n and

m can vary, i.e., the number of sides of the inscribed polygon

formed inside the circle. The values of n and m are chosen

randomly. Table II analyzes the value of the volume of a torus

estimated at 



= 3158.27341. Then we compare the value

of the PGT volume with the modeling, which converges when

it finds that the values of n and m increase.

Below is Fig. 2 of a torus and table II:

TABLE II. T

ORUS

IMULATION

Table II: Torus simulation

n m VPGT Absolute error Relative error

39 37 3143.12 15.15 0.4796929965

91 44 3147.55 10.72 0.3394263315

107 195 315.,73 0.54 0.01709796819

242 227 3157.87 0.40 0.01266516162

IV. N

UMERICAL

IMULATION

In this section, we will perform simulations using Geogebra

software and PGT theory to study the volume and surface area

of some solids designed for industrial use, such as packaging

and solids found in nature, such as eggs and pears.

A. Simulation in the packaging industry

It can also be shown that there are other objects used in in-

dustry, mainly in packaging, so we minimize the cost by op-

timizing the surface area and occupied volume with GeoGe-

bra software to perform the modeling and analysis. Figures 5

and 6 show two minimal surfaces used as the basis for the

production of some packaging, the catenoid, and the helicoid.

Numerical simulations are created by objects of geometric

models with mathematical concepts using computer graphics.

With its importance in reducing production time and costs,

with a potential impact on job creation and income, as seen in

[8]. Based on the use of software for manufacture of elec-

tronic components, we have used geometric modeling with

mathematics according to [10], where the PGT can be applied

to determine its volume and surface area in other products

used in the industry.

ISSN:1390-9266 e-ISSN:1390-9134 LAJC 2024

DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024

10.5281/zenodo. 10402067

F. Zastrow, L. Do Nascimento,

“Pappus-Guldin theorems applied to the study of solid modeling with GeoGebra software”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024

Fig.5. Catenoid

Fig. 6. Helicoid

Industry uses packaging to store its products, so it is im-

portant to study the format to be used. Reduce packaging

costs by paying attention to factors such as format; reduce

material costs. Among other things, analyzing these factors

can lead to reducing environmental impact and promoting

sustainable recycling. We can consider the use of minimal

surface areas in industrial packaging. The importance of stud-

ying packaging forms, which we refer to in the work of [7],

explores the feasibility and importance of new forms of PET

bottles. In this case, when designing new packaging, we

should primarily consider reducing the environmental impact

of packaging damage. The examples of these packages by de-

signers of PET bottle and yogurt figures are surfaces of revo-

lution that have an axis of rotation, but they are not minimal

surfaces. To be a minimal surface, it would have to have infi-

nite curves that fix two points on each edge.

B. Numerical models of packaging in the industry

Using real images of objects and using the resources of

the Geogebra software, we were able to model some packages

used in daily life.

Figures 7 to 13 present simulations considering objects mod-

eled by using splines where the PGT was applied.

Fig. 7 has the modeling of the PET bottle with the number

of steps n=150 and the value of the volume over 



=25.16

and the value over PGT of the area is 



=30.2.

Fig.7. PET bottle

Fig. 8 has the modeling of a yogurt container with number

of steps n = 150 and volume value over 



=78.75 and

value over PGT of the area is 



=62.98.

Fig. 9 has the modeling of coffee powder packaging as a

solid of revolution and a catenoid. The packaging is improved

by the designer to attract consumer attention and increase

sales through a naturally developed format.

From the visual point of view of packaging, the best

would be the catenoid, as ours is cheaper to produce and the

aluminum can has a strong construction and also for better

conservation.

Modeling of coffee powder packaging with the number

of steps n = 150 and volume value over 



= 48.59 and

value over PGT of the area is 



= 46.91.

Fig. 8. Yogurt container

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Fig. 9.

Coffee powder packaging

Fig. 10 has the modeling of a jar with the number of steps

n = 150 and the volume value over 



= 2.51 and value

over PGT of the area is 



= 16.19.

In industry, the packaging of chocolate powder looks like

a spiral, but it is not a spiral. This kind of filling cannot be

modeled with PGT because we cannot get a generating curve

around a fixed axis with the spline, in short, it is not a surface

of revolution.

Some packages are modified to achieve the best develop-

ment of a product to attract the public and sell more products

through advertising [2].

In the studied example, the chocolate milk can is not opti-

mized, and the production cost of can must have increased.

Fig. 10. Jar

Oblique cylindrical shapes are mainly used in the au-

tomotive industry, such as drills, screws, rotors, and helical

gears, because the material is more evenly distributed and

more resistant, which increases the quality of the product.

Fig. 11. Packaging of chocolate powder

We can model the packaging of chocolate powder if we

assume that it is obtained by a generation curve. To do this,

we can use the analysis of the model through PGT, as shown

in Fig. 11.

Thus, we obtain a modeling with a number of the steps n

= 150 and the volume value over 



=102.58 and the PGT

value of the area is 



=66.81.

Fig.12 has the modeling of an aluminum soda can with the

number of steps n = 150 and the volume value over 



=196.4 and the PGT value for the area is 



=111.86 .

Fig. 12. Aluminum soda can

Fig. 13 shows the modeling of an aluminum barrel with a

number of steps n = 150 and the volume value over 



95.65 and the PGT value for the area is 



=66.69.

Fig. 13. Aluminum barrel

C. Numerical models of packaging in nature

Figures 14 and 15 show the format of some packages

found in nature, such as pear and egg.

Fig. 14. Pear shape

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DOI:

LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 1, January 2024

10.5281/zenodo. 10402067

F. Zastrow, L. Do Nascimento,

“Pappus-Guldin theorems applied to the study of solid modeling with GeoGebra software”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024

Fig.14 shows the modeling of the shape of a pear with the

number of steps n = 150 and the volume value over 



=171.41 and the PGT value of the area is 



=104.39.

Fig. 15. Egg shape

In Fig. 15, we see the modeling of the shape of an egg

with the number of steps n = 150 and the volume value over





=106.15 and the PGT value of the area is 



=71.57.

It is important to analyze that the modeling of the pear and

the egg by PGT is possible because they are surfaces of rev-

olution which, since they occur in nature, have optimal shapes

for their volume and surface.

ONCLUSIONS

The modeling was done in free software GeoGebra. The

formats of some parcels were constructed by rotation curves

on the symmetry axis in both two and three dimensions. These

Pappus-Guldin Theorems (PGT) allow us to measure and an-

alyze the numerical value of these models, such as volume

and surface area. The advantage of the PGT method is that it

allows to analyze of numerical values of objects easily and

directly with Geogebra, the disadvantage of the method is that

it can be used only for solids obtained by rotation. The applied

program provides a simple language as long as the figure has

an axis of symmetry according to [14]. Since the work is ex-

tensive, we note that we combine theory and practice to pro-

vide practical results that can be applied in industries, such as

the packaging industry.

In future work, costs can be minimized, especially in the

production of packaging, by conducting a study on the use of

minimum surface area to determine the shapes of some pack-

aging, as well as a method to improve the use of packaging is

to reduce the material used in the construction of this packag-

ing and shaping to be able to reduce overall costs and bring

better transportation, storage, distribution, and consumption

conditions for the sale and consumption of goods. As a result,

logistics would spend less time to reduce costs, a strategy to

better serve customers, vehicles that consume less fuel, are

less harmful to the environment, and have potential in their

use of recycling in packaging.

ACKNOWLEDGEMENTS

This work is the result of research carried out at IMEF -

FURG (Institute of Mathematics, Statistics, and Physics of

the Federal University of Rio Grande).

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LATIN-AMERICAN JOURNAL OF COMPUTING (LAJC), Vol XI, Issue 2, July 2024

AUTHORS

Fernanda Zastrow Tavares, Brazilian, was born on April 9, 1992,

in the city of Rio Grande-RS and is currently a student of Applied

Mathematics at the Federal University of Rio Grande - FURG. She

entered academic life in the Mathematics program at the Federal

University of Rio Grande - FURG in 2010, graduated in 2016 and

entered the Applied Mathematics program in 2017, with the aim of

deepening her knowledge in the ﬁeld of applied mathematics. . Her

area of interest is algebra, algorithm analysis, numerical simulations,

mathematical modeling, and computer modeling in industry. In

2020, she began her studies in computer modeling, analyzing the

Pappus-Guldin theorem and its applications to surfaces of revolution

using Geogebra software and developing a monograph in applied

mathematics. In October 2022, she presented the result of her

research at the XXV National Meeting of Computational Modeling.

Luverci do Nascimento Ferreira, brazilian, born in Brasília-DF, is currently

an Adjunct Professor at the Institute of Mathematics, Statistics and

Physics of the Federal University of Rio Grande - IMEF-FURG, since

2008. He has a degree in Mathematics from the University of Brasília

- UnB ( 2003) and Master in Mathematics from the University of

Brasília – UnB (2006) in the area of Ordinary Dierential Equations.

PhD student in Computational Modeling at the Federal University of

Rio Grande since 2019, with a research area related to the study of

algorithms to solve fractional dierential equations using the statistical

method known as the Monte Carlo Method. He has experience in

teaching undergraduate courses in Mathematics, Applied Mathematics,

Mechanical Engineering and Business Mechanical Engineering and in

specialization courses for Mathematics Teachers. Research and interest

in the areas of Mathematical Analysis, Mathematical Epidemiology,

Biomathematics, Dynamic Geometry, Complex Variables, Ordinary and

Partial Dierential Equations, Fractional Calculus, Fractional Dierential

Equations, Stochastic Processes and Computational Modeling.

Fernanda Zastrow Tavares

Luverci do Nascimento Ferreira

F. Zastrow, L. Do Nascimento,

“Pappus-Guldin theorems applied to the study of solid modeling with GeoGebra software”,

Latin-American Journal of Computing (LAJC), vol. 11, no. 1, 2024