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

Pappus'un hacim teoremi, disk yöntemi gibi entegrasyon teknikleri kullanmadan, bir düzlem bölgesinin geometrisini dönme katısı hacmine bağlar.Teorem, bilinen bir merkeze sahip bir düzlem bölgesi bir eksen etrafında döndüğünde, ortaya çıkan hacmin bölgenin alanının dairesel yol boyunca merkez tarafından kat ettiği mesafenin çarpımı eşdeğer olduğunu belirtir.Örneğin, çörek gibi görünen üç boyutlu bir şekil olan torus olarak düşünelim. Düzleminde çemberle kesişmeyen bir çizgi etrafında r yarıçaplı bir çemberin döndürülmesiyle oluşur.Bu dairesel kesit, dönme sırasında katı oluşturur.Çemberin merkezi, dönme ekseninden R mesafesinde yer alır.Daire döndükçe, merkez kısmı çevresine eşit uzunluğunda dairesel bir yol izler.Teoremi kullanarak, torusun hacmi, kesit alanının centroid'in kat ettiği mesafeyle çarpılmasıyla bulunur.Pratik bir örnek, dar alanlarda yakıt depolamak için toroidal tankların kullanılmasıdır. Pappus Teoremi, alanı ve merkezin yol uzunluğunu kullanarak hacmi bulur; bu da mevcut yakıt kapasitesini göstermeye yardımcı olur.

pappus teoremi

Theorem of Pappus

Pappus Teoremi (Pappus&#x2013;Guldinus Teoremi olarak da bilinir), d&#xFC;zlemsel bir b&#xF6;lgenin veya d&#xFC;zlemsel bir e&#x11F;rinin d&#x131;&#x15F; bir eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;an cisimlerin hacmini ve y&#xFC;zey alan&#x131;n&#x131; belirlemek i&#xE7;in geometrik bir y&#xF6;ntem sunar. Teorem, birbiriyle ili&#x15F;kili iki &#xF6;nermeden olu&#x15F;ur. Birincisi, d&#xFC;zlemsel alanlar&#x131;n d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;an cisimlerin hacmini; ikincisi ise d&#xFC;zlemsel e&#x11F;rilerin d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;an y&#xFC;zey alan&#x131;n&#x131; ele al&#x131;r. Her iki sonu&#xE7; da, alan&#x131;n veya e&#x11F;rinin ortalama konumunu temsil eden geometrik merkezin konumuna ba&#x11F;l&#x131;d&#x131;r.Pappus&#x2019;un Birinci Teoremi, d&#xFC;zlemsel bir b&#xF6;lgenin, b&#xF6;lgeye d&#x131;&#x15F; bir eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;an cismin hacmi V&#x2019;nin, b&#xF6;lgenin alan&#x131; A ile geometrik merkezinin bir tam d&#xF6;n&#xFC;&#x15F; s&#x131;ras&#x131;nda kat etti&#x11F;i yolun &#xE7;arp&#x131;m&#x131;na e&#x15F;it oldu&#x11F;unu belirtir. Bu ili&#x15F;ki a&#x15F;a&#x11F;&#x131;daki gibi verilir:<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>Burada d, b&#xF6;lgenin geometrik merkezinden d&#xF6;nme eksenine olan dik uzakl&#x131;kt&#x131;r.&#xC7;ap&#x131; etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;len yar&#x131;&#xE7;ap&#x131; r olan bir yar&#x131;m daireyi ele alal&#x131;m. Yar&#x131;m dairenin alan&#x131; A = 1/2&#x3C0;r^2 ve geometrik merkezin &#xE7;apa olan uzakl&#x131;&#x11F;&#x131; d = 4r/3&#x3C0;&#x2019;dir. Bu de&#x11F;erler yerine konuldu&#x11F;unda elde edilir.<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>Bu sonu&#xE7; k&#xFC;renin hacmi i&#xE7;in klasik form&#xFC;le kar&#x15F;&#x131;l&#x131;k gelir.Pappus&#x2019;un &#x130;kinci Teoremi, d&#xFC;zlemsel bir e&#x11F;rinin d&#x131;&#x15F; bir eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;lmesiyle olu&#x15F;an y&#xFC;zey alan&#x131; S&#x2019;nin, e&#x11F;rinin yay uzunlu&#x11F;u L ile geometrik merkezinin bir tam d&#xF6;n&#xFC;&#x15F; s&#x131;ras&#x131;nda kat etti&#x11F;i yolun &#xE7;arp&#x131;m&#x131;na e&#x15F;it oldu&#x11F;unu belirtir. Bu ili&#x15F;ki &#x15F;u &#x15F;ekilde ifade edilir:<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>&#xD6;rnek olarak, merkezi d&#xF6;nme ekseninden R uzakl&#x131;kta bulunan, yar&#x131;&#xE7;ap&#x131; r olan bir daireyi ele alal&#x131;m. Bu dairenin eksen etraf&#x131;nda d&#xF6;nd&#xFC;r&#xFC;lmesi bir tor olu&#x15F;turur. Dairenin yay uzunlu&#x11F;u L = 2&#x3C0;r&#x2019;dir; geometrik merkez ise 2&#x3C0;R uzunlu&#x11F;unda dairesel bir yol izler. Teorem uyguland&#x131;&#x11F;&#x131;nda elde edilir.<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>Bu teoremler, geometrik merkez konumlar&#x131; ve geometrik &#xF6;l&#xE7;&#xFC;ler bilindi&#x11F;i s&#xFC;rece, do&#x11F;rudan integrasyon yapmaya gerek kalmadan hacimlerin ve y&#xFC;zey alanlar&#x131;n&#x131;n etkin bi&#xE7;imde hesaplanmas&#x131;na olanak tan&#x131;r. M&#xFC;hendislik ve uygulamal&#x131; bilimlerde Pappus&#x2013;Guldinus Teoremi, tanklar, kubbeler ve borular gibi d&#xF6;nel cisimlerin analizinde yayg&#x131;n olarak kullan&#x131;larak do&#x11F;ru tasar&#x131;m&#x131; ve malzeme miktar&#x131;n&#x131;n kestirimini destekler.

Pappus Teoremi

Pappus Teoremi (Pappus&#x2013;Guldinus Teoremi olarak da bilinir), d&#xFC;zlemsel bir b&#xF6;lgenin veya d&#xFC;zlemsel bir e&#x11F;rinin d&#x131;&#x15F; ...

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JoVE Core

Calculus

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Birim

Application of Techniques of Integration

İntegral Alma Tekniklerinin Uygulanması

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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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