Pappus centroid theorem pdf

How are these theorems proved without using calculus. Centroid of a triangle the centroid of a triangle is the point where the three medians coincide. Create marketing content that resonates with prezi video. To simplify notation, in what follows, in menelaus theorem we refer. In addition, pappus gave some apparently original results, such as the proposition that is commonly called pappus theorem involving a hexagon inscribed between two lines. A simple proof for the theorems of pascal and pappus marian palej geometry and engineering graphics centre, the silesian technical university of gliwice ul.

Finding surface area and volume of a sphere using only. The centroid of the area coincides with the center of symmetry. The concept of the first moment of an area is used to locate the centroid. The pappus guldin theorems suppose that a plane curve is rotated about an axis external to the curve. I did some calculation and found out the ratio of surface area and volume. Jan 22, 20 theorem of pappus tells us that volume is equal to area of the plane region, times the distance traveled by the centroid of the same plane region, if the plane region is revolved around the xaxis. Consider two straight lines emanating from point o and containing the points p 1 through p 6 as shown in the figure below. The centroid of an area is analogous to the center of gravity of a body. Now the second pappusguldin theorem gives the volume when this region is rotated through. Calculus 2 center of mass and pappus centroid theorem. If a plane area is rotated about an axis in its plane, but which does not cross the area, the volume swept out equals the area times the distance moved by the centroid. Pappus theorem article about pappus theorem by the. Symmetry if the region has an axis of symmetry, then the centroid.

The theorem, which can also be thought of as a generalization of the pythagorean theorem, is named after the greek mathematician pappus of alexandria 4th century ad, who discovered it. A centroid is easily visualized as the center of gravity or center of mass of a flat. Pappus s centroid theor em may refer to one of two theorems. In mathematics, pappuss centroid theorem is either of two related theorems dealing with the. The surface of revolution generated by a smooth curve. The proposition that the volume of a solid of revolution. Note that, by the symmetry, the centroid of the hexagon is 2.

The second pappus centroid theorem or the pappusguldin theorem states that the volume of a solid of revolution generated by rotating a plane region. This is precisely what pappus centroid theorem gives. Alternatively, given a mystic hexagon, the pappus con. These three points are the points of intersection of the opposite sides of the hexagon. They show that the second theorem you stated has a generalization when the circle that the centroid of f travels on is replaced with any sufficiently smooth simple closed space curve whose curvature never vanishes, however, the first theorem does not generalize easily. Let a be a region in the upper half plane with boundary curve c, let e be the solid of revolution formed. Request pdf the axiomatic destiny of the theorems of pappus and desargues we present the largely twentieth century history of the discovery of the significance of pappus and desargues for the. Area of surface of revolution the area of a surface of revolution is equal to the length of the generating curve multiplied by the distance traveled by the centroid of the curve while the surface is being genera. Pappus s centroid theorem, another theorem named for pappus of alexandria. Nothing is known of his life, other than what can be found in his own writings. Center of mass and centroids centroids of lines, areas, and volumes centroid is a geometrical property of a body when density of a body is uniform throughout, centroid and cm coincide dv v lines. Area of surface of revolution the area of a surface of revolution is equal to the length of the generating curve multiplied by the distance traveled by the centroid of the curve while the surface is being generated.

Suppose r is revolved about the line l which does not cut. Solid of rotation, pappus centroid theorem a solid of rotation is the figure that results from rotating a plane figure about an external axis an axis on the same plane as the figure such that no two points of the figure are on opposite sides of the axis. Pappus s first theorem states that the area of a surface generated by rotating a figure about an external axis a distance from its centroid equals the product of the arc length of the generating figure and the distance traversed by. Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x 2. By pappus theorem the volume generated by revolving dabout the xaxis is 2a. An analytic proof of the theorems of pappus and desargues. Pappus s theorem, in mathematics, theorem named for the 4thcentury greek geometer pappus of alexandria that describes the volume of a solid, obtained by revolving a plane region d about a line l not intersecting d, as the product of the area of d and the length of the circular path traversed by the centroid of d during the revolution. Jul 07, 2016 pappus s centroid theorems were discovered 17 centuries ago, when calculus wasnt invented yet. The surface of revolutiongenerated bya smooth curve. Theorems of pappus and goldinus mechanical engineering notes. In mathematics, pappus s hexagon theorem attributed to pappus of alexandria states that given one set of collinear points,, and another set of collinear points,, then the intersection points, of line pairs and, and, and are collinear, lying on the pappus line. The theorems are attributed to pappus of alexandria and paul guldin. In mathematics, pappuss centroid theorem also known as the guldinus theorem, pappusguldinus theorem or pappuss theorem is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.

I wonder if it is possible to derive surface area and volume of a sphere seperately using techniques involving pappus theorem. What does pappus mean definition of pappus word finder. If the pappus line u \displaystyle u and the lines g, h \displaystyle g,h have a point in common, one gets the socalled little version of pappus s theorem 2. For deriving the second pappus s centroid theorem, we suppose that the region defined by a. Theorem of pappus to find volume of revolution calculus 2. Pappus flower structure, a structure within certain flowers. Pappuss centroid theorem volume by george kotzabassis on prezi. An application of pappus involution theorem in euclidean and noneuclidean geometry. May 24, 2014 if youd like to make a donation to support my efforts look for the tip the teacher button on my channels homepage. There are two results of pappus which relate the centroids to surfaces and solids of revolutions. Theorems of pappus on surfaces of revolution wolfram. Theorem of pappus to find volume using the centroid. Generalizations of pappus centroid theorem via stokes theorem.

The theorem of pascal concerning a hexagon inscribed in a conic. An application of pappus involution theorem in euclidean and. Pappuss theorem also known as pappuss centroid theorem, pappusguldinus theorem or the guldinus theorem deals with the areas of surfaces of revolution. Of course, this does not make the computation trivial in general, since computing the centroid of a region or curve is not easy, even for relatively simple shapes. Pappus centroid theorem pdf the surface of revolution generated by a smooth curve. Nothing is known of his life, other than, from his own writings that he had a son named hermodorus, and was a. Aug 25, 2015 there are two theorems, both saying similar things. A bridge between algebra and geometry article pdf available in the american mathematical monthly 1096 june 2002 with 2,653 reads how we measure reads. His great work a mathematical collection is an important source of information about ancient greek mathematics. The first theorem of pappus states that the surface area sof a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance d 1 traveled by the curves geometric centroid. May 12, 2018 homework statement determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices 1,0, 0,0, 1,1. The surface area of a solid of revolution is the arc length of the generating curve multiplied by the distance traveled by the centroid of the curve. Centroid is a geometrical property of a body when density of a body is uniform throughout, centroid and cm coincide dv v lines.

In applying menelaus theorem, we need to identify a trianlge and three collinear points respectively on its sides. Prove pappuss centroid theorems without calculus physics. The following table summarizes the surface areas calculated. Profrobbob i introduce the theorem of pappus and then work. Article information source involve, volume 8, number 5 2015, 771785. Media in category pappus guldinus theorem the following 6 files are in this category, out of 6 total. A simple proof for the theorems of pascal and pappus.

Dec 22, 2019 applications of the theorems of pappus. If one restricts the projective plane such that the pappus line is the line at infinity, one gets the affine version of pappus s theorem shown in the second diagram. Pappus of alexandria greek mathematician britannica. An area is symmetric with respect to a center o if for every element da at x,y there exists an area da of equal area at x,y. Century ad proposed two theorems for determining the area and volume of surfaces of revolution. Nine proofs and three variations x y z a b c a b z y c x b a z x c y fig. Pappus centroid theorem pdf pappus centroid theorem pdf pappus centroid theorem pdf download.

Using calculus, the centroid of the region bounded by the curve y fx and the xaxis in the interval a,b has x and y coordinates x 1 a z b a xfxdx and y 1 a z b a 1 2 fx 2 dx, where a is the area of the region. After this the point comes back from a very far position on. Use pappus s theorem for surface area and the fact that the surface area of a sphere of radius q is 4piq2 to find the centroid of the semicircle yq2x20. Pappus s centroid theorem may refer to one of two theorems. Areas of surfaces of revolution, pappuss theorems iitk. These theorems enable us to work out the volume of a solid of revolution if we know the position of the centroid of a plane area, or vice versa. Pappuss theorem, in mathematics, theorem named for the 4thcentury greek geometer pappus of alexandria that describes the volume of a solid, obtained by revolving a plane region d about a line l not intersecting d, as the product of the area of d and the length of the circular path traversed by the centroid of d during the revolution. Lesson 55 centroid theorem of pappus guldinus volume and surface. In this article w egiv an analytic proofpappus theorem and. The axiomatic destiny of the theorems of pappus and. Apply the theorem of pappus guldinus to evaluate the volumes or revolution for the rectangular rim section and the inner cutout section.

Finding surface area and volume of a sphere using only pappus. This paper provides a novel proof of a generalization of pappus centroid theorem on ndimensional tubes using stokes theorem on manifolds. Much of it is overwhelming, and some of it is wrong. In the situation with zero slope both lines are parallel and the intersection point vanishes. To interpret the explanations on or computation meets knowledge you need to know what a centroid is. Pappus s area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. Theorems of pappus and goldinus mechanical engineering. When the six points are ordered as a, f, b, d, c, f the resulting polygon is just pascals mystic hexagon. Body with small but constant thickness t crosssectional area a. The first theorem of pappus states that the surface area s of a surface of revolution generated by the. Now towards the end of our last lecture, we had started with theorems of pappus guldinus. On menelaus theorem singapore mathematical society. The usual pappus theorem is just the situation whereby the conic degenerates into a pair of lines.