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

Pappus&#8217;s Theorem for volume connects the geometry of a plane region with the volume of its solid of revolution, without using integration techniques like the disk method.The theorem states that when a plane region with a known centroid rotates around an axis, the resulting volume equals the region's area times the distance traveled by its centroid along the circular path.For example, consider a three-dimensional shape that looks like a doughnut, called a torus. It is formed by rotating a circle of radius r around a line in its plane that does not intersect the circle.This circular cross-section generates the solid during rotation.The circle&#8217;s centroid lies at a distance R from the rotation axis.As the circle rotates, its centroid follows a circular path with a length equal to its circumference.Using the theorem, the volume of the torus is found by multiplying the cross-sectional area by the distance traveled by the centroid.A practical example is the use of toroidal tanks to store fuel in tight spaces. Pappus&#8217;s Theorem uses the area and the centroid&#8217;s path length to find the volume, which helps show the available fuel capacity.

stelling van pappus

Theorem of Pappus

De stelling van Pappus, ook bekend als de stelling van Pappus&#x2013;Guldinus, biedt een geometrische methode om het volume en de oppervlakte te bepalen van lichamen die ontstaan door de rotatie van een vlak gebied of een vlakke kromme rond een externe as. De stelling bestaat uit twee samenhangende beweringen. De eerste heeft betrekking op het volume van lichamen die worden gevormd door de rotatie van vlakke gebieden, terwijl de tweede betrekking heeft op de oppervlakte die ontstaat door de rotatie van vlakke krommen. Beide resultaten zijn afhankelijk van de ligging van het centro&#xEF;de, of zwaartepunt, dat de gemiddelde positie van het gebied of de kromme weergeeft.De eerste stelling van Pappus luidt dat het volume V van een lichaam dat ontstaat door de rotatie van een vlak gebied om een as buiten dat gebied gelijk is aan het product van de oppervlakte A van het gebied en de afstand die het zwaartepunt tijdens &#xE9;&#xE9;n volledige omwenteling aflegt. Deze relatie wordt gegeven door<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>waarbij d de loodrechte afstand is van het zwaartepunt van het gebied tot de rotatieas.Beschouw een halve cirkel met straal r die rond haar diameter wordt geroteerd. De oppervlakte van de halve cirkel is A = 1/2&#x3C0;r^2, en de afstand van het zwaartepunt tot de diameter is d = 4r/3&#x3C0;. Door deze waarden te substitueren verkrijgt men<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>hetgeen overeenkomt met de klassieke formule voor het volume van een bol.De tweede stelling van Pappus luidt dat de oppervlakte S die ontstaat door de rotatie van een vlakke kromme om een externe as gelijk is aan de booglengte L van de kromme vermenigvuldigd met de afstand die het zwaartepunt aflegt. Dit wordt uitgedrukt als<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>Als voorbeeld beschouw een cirkel met straal r waarvan het middelpunt zich op een afstand R van de rotatieas bevindt. Door deze cirkel om de as te roteren ontstaat een torus. De booglengte van de cirkel is L = 2&#x3C0;r, en het zwaartepunt beschrijft een cirkelvormig pad van lengte 2&#x3C0;R. Door de stelling toe te passen verkrijgt men<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>Deze stellingen maken het mogelijk om volumes en oppervlakken effici&#xEB;nt te berekenen zonder expliciet te integreren, mits de posities van de zwaartepunten en de relevante geometrische grootheden bekend zijn. In de ingenieurswetenschappen en toegepaste wetenschappen wordt de stelling van Pappus&#x2013;Guldinus veelvuldig toegepast bij de analyse van rotatielichamen, zoals tanks, koepels en pijpen, ter ondersteuning van nauwkeurig ontwerp en nauwkeurige materiaalraming.

Stelling van Pappus

De stelling van Pappus, ook bekend als de stelling van Pappus&#x2013;Guldinus, biedt een geometrische methode om het volume en de oppervlakte te bepalen ...

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Calculus

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Application of Techniques of Integration

Toepassing van integratietechnieken

KEY TERMS AND CONCEPTS

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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

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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