Name: pappus theorem for volume and surface area
Uploaded: 2026-01-12T12:24:33.000Z
Duration: 1 min 24 s
Channel: JoVE

Il teorema di Papp per il volume collega la geometria di una regione piana con il volume del suo solido di rivoluzione, senza utilizzare tecniche di integrazione come il metodo del disco.Il teorema afferma che quando una regione piana con un barichio noto ruota attorno a un asse, il volume risultante è uguale all'area della regione moltiplicata per la distanza percorsa dal suo baricentroide lungo il percorso circolare.Ad esempio, consideriamo una forma tridimensionale che assomiglia a una ciambella, chiamata toro. Si forma ruotando un cerchio di raggio r attorno a una retta nel suo piano che non interseca il cerchio.Questa sezione trasversale circolare genera il solido durante la rotazione.Il centroide del cerchio si trova a una distanza R dall'asse di rotazione.Mentre il cerchio ruota, il suo barifactori segue un percorso circolare con una lunghezza pari alla sua circonferenza.Usando il teorema, il volume del toro si determina moltiplicando l'area della sezione trasversale per la distanza percorsa dal baricentro.Un esempio pratico è l'uso di serbatoi toroidali per immagazzinare carburante in spazi ristretti. Il teorema di Pappus utilizza l'area e la lunghezza del percorso del baricentrico per trovare il volume, il che aiuta a mostrare la capacità di combustibile disponibile.

teorema di pappo

Theorem of Pappus

Il teorema di Pappo, noto anche come teorema di Pappo-Guldino, fornisce un metodo geometrico per determinare il volume e l&#x2019;area superficiale dei solidi generati dalla rotazione di una regione piana o di una curva piana attorno a un asse esterno. Il teorema consta di due enunciati fra loro correlati. Il primo riguarda il volume dei solidi ottenuti ruotando aree piane, mentre il secondo riguarda l&#x2019;area superficiale generata dalla rotazione di curve piane. Entrambi i risultati dipendono dalla posizione del centroide, che rappresenta la posizione media dell&#x2019;area o della curva.Il primo teorema di Pappo afferma che il volume V di un solido formato dalla rotazione di una regione piana attorno a un asse esterno alla regione &#xE8; uguale al prodotto dell&#x2019;area A della regione e della distanza percorsa dal suo centroide durante una rivoluzione completa. Questa relazione &#xE8; data da<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/40592/40592_Equation_1_editor_40592.svg" alt="40592_Equation_1.svg" /></figure>dove d &#xE8; la distanza perpendicolare tra il centroide della regione e l&#x2019;asse di rotazione.Si consideri un semicerchio di raggio r ruotato attorno al diametro. L&#x2019;area del semicerchio &#xE8; A = 1/2&#x3C0;r^2 e la distanza del centroide dal diametro &#xE8; d = 4r/3&#x3C0;. Sostituendo questi valori si ottiene<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/40592/40592_Equation_2_editor_40592.svg" alt="40592_Equation_2.svg" /></figure>che corrisponde alla formula classica per il volume di una sfera.Il secondo teorema di Pappo afferma che l&#x2019;area superficiale S generata dalla rotazione di una curva piana attorno a un asse esterno &#xE8; uguale alla lunghezza d&#x2019;arco L della curva moltiplicata per la distanza percorsa dal suo centroide. Tale relazione &#xE8; espressa da<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/40592/40592_Equation_3_editor_40592.svg" alt="40592_Equation_3.svg" /></figure>A titolo di esempio, si consideri una circonferenza di raggio r il cui centro si trova a una distanza R dall&#x2019;asse di rotazione. La rotazione di questa circonferenza attorno all&#x2019;asse produce un toro. La lunghezza d&#x2019;arco della circonferenza &#xE8; L = 2&#x3C0;r e il centroide percorre una traiettoria circolare di lunghezza 2&#x3C0;R. Applicando il teorema, si ottiene<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/24;" src="https://cloudfront.jove.com/files/ftp_upload/40592/40592_Equation_4_editor_40592.svg" alt="40592_Equation_4.svg" /></figure>Questi teoremi consentono di calcolare in modo efficiente volumi e aree superficiali senza ricorrere a un&#x2019;integrazione esplicita, purch&#xE9; siano note le posizioni dei centroidi e le misure geometriche. In ingegneria e nelle scienze applicate, il teorema di Pappo-Guldino &#xE8; ampiamente impiegato nell&#x2019;analisi dei solidi di rivoluzione, quali serbatoi, cupole e tubazioni, a sostegno di una progettazione accurata e di stime affidabili dei materiali.

Teorema di Pappo

Il teorema di Pappo, noto anche come teorema di Pappo-Guldino, fornisce un metodo geometrico per determinare il volume e l&#x2019;area superficiale dei ...

Education

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JoVE Core

Calculus

Analisi matematica

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Unità

Application of Techniques of Integration

Applicazione delle tecniche di integrazione

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

pappus theorem for volume and surface area

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis.

The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled by its centroid in one full turn. The distance d is the perpendicular distance from the centroid to the axis of rotation.

A semicircle shows how the first theorem works. For a semicircle of radius r, the area is 1/2 r^2. Its centroid is 4r/3 from the diameter. When these values are used, the result matches the classical formula for the volume of a sphere.

The second part of the theorem gives the surface area made by rotating a plane curve about an external axis. Here, the surface area equals the arc length of the curve multiplied by the distance traveled by its centroid. The arc length is the length of the curve, and the centroid again follows a circular path during rotation.

A circle provides the second example. For a circle of radius r whose center is a distance R from the axis, rotation forms a torus. The circle has arc length 2r, and its centroid travels a path of length 2R. Using the theorem gives the surface area of the torus.

These theorems help students and engineers find volumes and surface areas without direct integration, as long as the centroid location and geometric measurements are known. They are useful in applied science for shapes such as tanks, domes, and pipes. That makes the Pappus–Guldinus Theorem a practical tool for design and material estimation.

Pappus Theorem for Volume and Surface Area

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theorem of pappus

Pappus Theorem connects centroid motion to the volume and surface area of solids of revolution. It is also called the Pappus–Guldinus Theorem. The method uses geometry to find these values when a plane region or a plane curve is rotated about an external axis. The theorem has two related parts. The first part gives the volume of a solid formed by rotating a plane region around an axis outside the region. In this case, the volume equals the area of the region multiplied by the distance traveled...

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key terms and concepts

Learn how the Pappus–Guldinus Theorem uses centroid motion to find the volume of solids and surface area of rotated curves, with sphere and torus examples.

Pappus Theorem for Volume and Surface Area | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Pappus Theorem for Volume and Surface Area

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