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

Pappus' Satz für das Volumen verbindet die Geometrie eines ebenen Bereichs mit dem Volumen seines Rotationskörpers, ohne Integrationstechniken wie die Scheibenmethode anzuwenden.Der Satz besagt, dass, wenn sich ein ebener Bereich mit einem bekannten Schwerpunkt um eine Achse rotiert, das resultierende Volumen der Fläche des Bereichs multipliziert mit der Distanz, die sein Schwerpunkt entlang des kreisförmigen Weges zurückgelegt hat.Betrachten wir zum Beispiel eine dreidimensionale Form, die wie ein Donut aussieht, genannt Torus. Sie entsteht, indem ein Kreis mit Radius r um eine Gerade in ihrer Ebene gedreht wird, die den Kreis nicht schneidet.Dieser kreisförmige Querschnitt erzeugt den Festkörper während der Rotation.Der Schwerpunkt des Kreises liegt in einem Abstand R von der Rotationsachse.Während sich der Kreis rotiert, folgt sein Schwerpunkt einem kreisförmigen Weg mit einer Länge, die seinem Umfang entspricht.Mit Hilfe des Theorems wird das Volumen des Torus ermittelt, indem man die Querschnittsfläche mit der vom Schwerpunkt zurückgelegten Distanz multipliziert.Ein praktisches Beispiel ist die Verwendung von toroidalen Tanks zur Lagerung von Treibstoff in engen Räumen. Der Satz von Pappus verwendet die Fläche und die Weglänge des Schwerpunkts, um das Volumen zu bestimmen, was hilft, die verfügbare Treibstoffkapazität darzustellen.

satz von pappus

Theorem of Pappus

Der Satz von Pappus, auch bekannt als Satz von Pappus&#x2013;Guldinus, bietet eine geometrische Methode zur Bestimmung von Volumen und Oberfl&#xE4;che von Rotationsk&#xF6;rpern, die durch die Drehung einer ebenen Fl&#xE4;che oder einer ebenen Kurve um eine &#xE4;u&#xDF;ere Achse entstehen. Der Satz besteht aus zwei miteinander verbundenen Aussagen. Die erste behandelt das Volumen von K&#xF6;rpern, die durch die Rotation ebener Fl&#xE4;chen entstehen, die zweite die Oberfl&#xE4;che, die durch die Rotation ebener Kurven erzeugt wird. Beide Ergebnisse h&#xE4;ngen von der Lage des Schwerpunkts der Fl&#xE4;che beziehungsweise der Kurve ab, der die mittlere Lage der Fl&#xE4;che oder Kurve repr&#xE4;sentiert.Der erste Satz von Pappus besagt, dass das Volumen V eines Rotationsk&#xF6;rpers, der durch die Drehung einer ebenen Fl&#xE4;che um eine Achse au&#xDF;erhalb dieser Fl&#xE4;che entsteht, gleich dem Produkt aus der Fl&#xE4;che A und der vom Schwerpunkt w&#xE4;hrend einer vollst&#xE4;ndigen Umdrehung zur&#xFC;ckgelegten Wegstrecke ist. Diese Beziehung wird ausgedr&#xFC;ckt durch:<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>wobei d der senkrechte Abstand des Schwerpunkts der Fl&#xE4;che zur Rotationsachse ist.Betrachtet wird ein Halbkreis mit dem Radius r, der um seinen Durchmesser rotiert wird. Die Fl&#xE4;che des Halbkreises betr&#xE4;gt A = 1/2&#x3C0;r^2, und der Abstand seines Schwerpunkts zum Durchmesser ist d = 4r/3&#x3C0;. Durch Einsetzen dieser Werte erh&#xE4;lt man:<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>was der klassischen Formel f&#xFC;r das Volumen einer Kugel entspricht.Der zweite Satz von Pappus besagt, dass die Oberfl&#xE4;che S, die durch die Drehung einer ebenen Kurve um eine &#xE4;u&#xDF;ere Achse entsteht, gleich der Bogenl&#xE4;nge L der Kurve multipliziert mit der vom Schwerpunkt zur&#xFC;ckgelegten Wegstrecke ist. Dies wird ausgedr&#xFC;ckt durch:<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>Beispielsweise wird ein Kreis mit Radius r betrachtet, dessen Mittelpunkt im Abstand R von der Rotationsachse liegt. Die Drehung dieses Kreises um die Achse erzeugt einen Torus. Die Bogenl&#xE4;nge des Kreises betr&#xE4;gt L = 2&#x3C0;r, und der Schwerpunkt durchl&#xE4;uft eine Kreisbahn der L&#xE4;nge 2&#x3C0;R. Die Anwendung des Satzes liefert:<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>Diese S&#xE4;tze erm&#xF6;glichen eine effiziente Bestimmung von Volumina und Oberfl&#xE4;chen ohne explizite Integration, vorausgesetzt, die Lage der Schwerpunkte und die geometrischen Ma&#xDF;e sind bekannt. In den Ingenieur- und Anwendungswissenschaften findet der Satz von Pappus&#x2013;Guldinus breite Anwendung bei der Analyse von Rotationsk&#xF6;rpern wie Tanks, Kuppeln und Rohren und unterst&#xFC;tzt eine pr&#xE4;zise Konstruktion sowie Materialabsch&#xE4;tzung.

Satz von Pappus

Der Satz von Pappus, auch bekannt als Satz von Pappus&#x2013;Guldinus, bietet eine geometrische Methode zur Bestimmung von Volumen und Oberfl&#xE4;che von ...

Education

Lehre

JoVE Core

Calculus

Analysis

Unit

Einheit

Application of Techniques of Integration

Anwendung von Integrationstechniken

KEY TERMS AND CONCEPTS

WICHTIGE BEGRIFFE UND KONZEPTE

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

pappus-theorem-for-volume-and-surface-area

arc length of a curve

Measuring Curved Paths with Calculus

Measuring curved paths with calculus is important in engineering projects. Roller coaster tracks, cables, and railway lines often follow smooth curves instead of straight lines. Accurate path length helps with planning and design. A curve is first split into small segments. Each short segment is treated as a straight line between two nearby points on the curve. Adding these line segments gives an estimate of the total distance along the path. The estimate becomes better when the segments are...

arc length of a curve: problem solving

Catenary Cable Length in Power Lines

Catenary cable length helps engineers measure the full span of a hanging power line. A high-voltage line stretches 40 meters between two transmission towers and sags 10 meters because of gravity and thermal expansion. The cable forms a catenary, which is the curve of a uniform, flexible cable under its own weight. A catenary is not the same as a parabola. It is described by the hyperbolic cosine function, which gives a more accurate shape for a suspended cable. In this setup, engineers use the...

arc length function

Measuring Distance Along a Curve

The arc length function measures the distance traveled along a smooth curve from a fixed starting point to a moving endpoint. It is useful when a straight-line estimate is not accurate enough. For curves that are continuous and differentiable, arc length gives a precise way to track length along the curve. To find this distance, the curve is broken into many small pieces. Each piece is treated like a short straight line. Its length depends on the horizontal change and the vertical change over...

area of a surface of revolution

Calculating Surface Area from Rotation

Surfaces of revolution are created when a two-dimensional curve is rotated around an axis. The rotation produces a three-dimensional shape. In engineering, this calculation can help determine the surface area of a rocket nozzle. That information matters when a heat-resistant coating must be applied evenly. When a curve is revolved about the x-axis, it sweeps out a continuous surface. To find its area, the curve is first broken into small straight-line pieces over short intervals. Each rotated...

applications of integration to find hydrostatic pressure

applications-of-integration-to-find-hydrostatic-pressure

Hydrostatic Force on Dam Walls with Integration

Hydrostatic force is the total force a fluid at rest exerts on a surface. For a horizontal surface at a fixed depth, the pressure is constant. It is found by multiplying fluid density, gravitational acceleration, and depth. A vertical dam wall is different because pressure increases with depth. The water above each point adds to the pressure below it. That means the force is not spread evenly across the wall. Integration helps measure this changing force. Pressure in a still fluid rises...

applications of integration to find centers of mass

applications-of-integration-to-find-centers-of-mass

Center of Mass in Discrete and Continuous Systems

Center of mass can be found from rotational equilibrium and from integration. A balanced plank on a pivot shows the idea clearly. When the net torque about the pivot is zero, the system is in balance. Torque is the force on an object times its perpendicular distance from the pivot. At that point, the pivot lies at the center of mass. For a set of discrete point masses, the center of mass is the position where the masses balance by weight. Each mass contributes its mass times its position,...

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...

applications of integration to find consumer surplus

applications-of-integration-to-find-consumer-surplus

Consumer Surplus from Demand and Integration

Consumer surplus measures the economic gain buyers receive when they pay less than the highest price they are willing to pay. In microeconomics, this idea comes from the demand function, which links quantity to the price consumers will pay. Because willingness to pay usually falls as quantity rises, the demand curve slopes downward. When a product is sold at a fixed price, some buyers would have paid more for the first units they bought. The difference between that willingness to pay and the...

applications of integration to find blood flow

applications-of-integration-to-find-blood-flow

Using Integration to Model Blood Flow

Blood flow in a cylindrical blood vessel can be modeled with integration and laminar flow. Laminar flow is smooth motion in parallel layers. In this model, blood speed changes with distance from the center of the vessel. The blood moves fastest along the central axis. As the distance from the center increases, the velocity decreases. At the vessel wall, the velocity drops to zero because of viscous drag. This changing speed across the cross-section is the key idea behind the calculation. To...

introduction to normal distributions

Normal Distribution in Test Scores

Normal distribution helps explain how standardized test scores can be spread across a group. It is a key idea in statistics for describing symmetry, center, and variability in large sets of data. Test score data often start as a histogram. In a histogram, each bar shows how many scores fall within a score range. To turn that count-based graph into a continuous probability model, the histogram is normalized by dividing each bar height by the total number of observations and by the bin width. <br...

applications of integration to probability density functions

applications-of-integration-to-probability-density-functions

Finding Probability with Area Under a PDF

Continuous probability distributions describe random variables that can take any real value within a set range. Height is one example, since it can be measured with growing precision, such as 163.5 or 165.25 centimeters. That makes height a continuous random variable. The pattern of a continuous random variable is shown with a probability density function, or PDF. A PDF shows how probability is spread across the possible values. In a continuous distribution, probabilities are assigned to...

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

Description

Transcript

Tags