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

El Teorema de Pappus para el volumen conecta la geometría de una región plana con el volumen de su sólido de revolución, sin utilizar técnicas de integración como el método del disco.El teorema establece que cuando una región plana con un centroide conocido gira alrededor de un eje, el volumen resultante es igual al área de la región multiplicada por la distancia recorrida por su centroide a lo largo del camino circular.Por ejemplo, considera una forma tridimensional que se parece a un donut, llamada toro. Se forma girando un círculo de radio r alrededor de una línea en su plano que no intersecta el círculo.Esta sección transversal circular genera el sólido durante la rotación.El centroide del círculo se encuentra a una distancia R del eje de rotación.A medida que el círculo gira, su centroide sigue un recorrido circular con una longitud igual a su circunferencia.Usando el teorema, el volumen del toro se determina multiplicando el área de la sección transversal por la distancia recorrida por el centroide.Un ejemplo práctico es el uso de tanques toroidales para almacenar combustible en espacios reducidos. El teorema de Pampus utiliza el área y la longitud del recorrido del centroide para encontrar el volumen, lo que ayuda a mostrar la capacidad de combustible disponible.

teorema de pappus

Theorem of Pappus

El teorema de Pappus, tambi&#xE9;n conocido como el teorema de Pappus&#x2013;Guldinus, proporciona un m&#xE9;todo geom&#xE9;trico para determinar el volumen y el &#xE1;rea de superficie de los s&#xF3;lidos generados por la revoluci&#xF3;n de una regi&#xF3;n plana o una curva plana alrededor de un eje externo. El teorema consta de dos enunciados relacionados. El primero aborda el volumen de los s&#xF3;lidos formados al rotar &#xE1;reas planas, mientras que el segundo aborda el &#xE1;rea de superficie generada al rotar curvas planas. Ambos resultados dependen de la ubicaci&#xF3;n del centroide, que representa la posici&#xF3;n media del &#xE1;rea o de la curva.El primer teorema de Pappus establece que el volumen V de un s&#xF3;lido formado al rotar una regi&#xF3;n plana alrededor de un eje externo a dicha regi&#xF3;n es igual al producto del &#xE1;rea A de la regi&#xF3;n por la distancia recorrida por su centroide durante una revoluci&#xF3;n completa. Esta relaci&#xF3;n se expresa como:<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>donde d es la distancia perpendicular desde el centroide de la regi&#xF3;n hasta el eje de rotaci&#xF3;n.Consid&#xE9;rese un semic&#xED;rculo de radio r que gira alrededor de su di&#xE1;metro. El &#xE1;rea del semic&#xED;rculo es A = 1/2&#x3C0;r^2, y la distancia desde su centroide hasta el di&#xE1;metro es d = 4r/3&#x3C0;. Sustituyendo estos valores, se obtiene:<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>que corresponde a la f&#xF3;rmula cl&#xE1;sica para el volumen de una esfera.El segundo teorema de Pappus establece que el &#xE1;rea de superficie S generada al rotar una curva plana alrededor de un eje externo es igual a la longitud del arco L de la curva multiplicada por la distancia recorrida por su centroide. Esto se expresa como:<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 modo de ejemplo, considere un c&#xED;rculo de radio r cuyo centro se halla a una distancia R del eje de rotaci&#xF3;n. Al rotar este c&#xED;rculo alrededor del eje, se obtiene un toro. La longitud del arco del c&#xED;rculo es L = 2&#x3C0;r, y el centroide recorre una trayectoria circular de longitud 2&#x3C0;R. La aplicaci&#xF3;n del teorema da como resultado:<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>Estos teoremas permiten el c&#xE1;lculo eficiente de vol&#xFA;menes y &#xE1;reas de superficie sin integraci&#xF3;n expl&#xED;cita, siempre que se conozcan las posiciones de los centroides y las medidas geom&#xE9;tricas. En ingenier&#xED;a y ciencias aplicadas, el teorema de Pappus&#x2013;Guldinus se utiliza ampliamente en el an&#xE1;lisis de s&#xF3;lidos de revoluci&#xF3;n, como tanques, c&#xFA;pulas y tuber&#xED;as, lo que permite un dise&#xF1;o preciso y la estimaci&#xF3;n adecuada de materiales.

Teorema de Pappus

El teorema de Pappus, tambi&#xE9;n conocido como el teorema de Pappus&#x2013;Guldinus, proporciona un m&#xE9;todo geom&#xE9;trico para determinar el ...

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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

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calculus

Pappus Theorem for Volume and Surface Area

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