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**Liceo Scientifico Isaac Newton**

Maths course Solid of revolution Professor Tiziana De Santis Read by Cinzia Cetraro

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A solid of revolution is obtained from the rotation of a plane figure around a straight line r, the axis of rotation; if the rotation angle is 360° we have a complete rotation All points P of the plane figure describe a circle belonging to the plane that is perpendicular to the axis and passing through the point P axis P P r

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Cylinder The infinite cylinder is the part of space obtained from the complete rotation of a straight line s around a parallel straight line r The part of an infinite cylinder delimited by two parallel planes is called a cylinder, if these planes are perpendicular to the rotation axis, then it is called a right cylinder r s r s s – generatrix r – axis

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**The sides perpendicular to the height are called radii of base**

The cylinder is also obtained from the rotation of a rectangle around one of its sides It is called height The sides perpendicular to the height are called radii of base The bases of the cylinder are obtained from the complete rotation of the radii of the base base height base radius

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**If we consider a half-line s having V as the initial point**

Cone If we consider a half-line s having V as the initial point The half-line s describes an infinite conical surface and the point V is called vertex of the cone and a straight line r passing through V called axis the infinite cone is the part of space obtained from the complete rotation of the angle α around r V V α α s r s r infinite cone infinite conical surface

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If the infinite cone is intersected by a plane perpendicular to the axis of rotation, the portion of the solid bounded between the plane and the vertex is called right circular cone The right circular cone is also obtained from the rotation of a right triangle around one of its catheti A cone is called equilateral if its apothem is congruent to the diameter of the base V base H P VH - height VP - apothem HP - radius of base

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**a small cone that is similar to the previous one and a truncated cone**

If we section a cone with a plane that is parallel to the base, we obtain two solids: a small cone that is similar to the previous one and a truncated cone H α V α’ H’ small cone truncated cone

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**Hp: α // α’ VH ┴ α Th: C : C’ = VH2 : VH’ 2**

Theorem: the measure of the areas C and C’, obtained from a parallel section, are in proportion with the square of their respective heigths H α V α’ H’ C C’ Hp: α // α’ VH ┴ α Th: C : C’ = VH2 : VH’ 2

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Sphere A spheric surface is the boundary formed by the complete rotation of a half-circumference around its diameter The rotation of a half-circle generate a solid, the sphere The centre of the half-circle is the center of the sphere, while its radius is the distance between all points on the surface and the centre The sphere is completely symmetrical around its centre called symmetry centre Every plane passing through the centre of a sphere is a symmetry plane The straight-lines passing through its centre are symmetry axes PC - radius C C - center P

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**Positions of a straight line in relation to a spheric surface**

B A A Secant: d < r Tangent: d = r External: d > r d - distance from centre C to straight line s r - radius of the sphere

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**Position of a plane in relation to a spheric surface**

TANGENT PLANE: intersection is a point SECANT PLANE: intersection is a circle EXTERNAL PLANE: no intersection

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Torus The torus is a surface generated by the complete rotation of a circle around an external axis s coplanar with the circle s s

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**Surface area and volume calculus**

Habakkuk Guldin (1577 –1643)

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**Pappus-Guldin’s Centroid Theorem**

Surface area calculus Pappus-Guldin’s Centroid Theorem The measure of the area of the surface generated by the rotation of an arc of a curve around an axis, is equal to the product between the length l of the arc and the measure of the circumference described by its geometric centroid (2 π d ) S = 2 π d l

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**Cone l=√h2+r2 d =r/2 Cylinder l=h d=r SL=π r √ h2+ r2 SL=2 π r h l**

Geometric centroid l=√h2+r2 d =r/2 SL=π r √ h2+ r2 Cylinder h r l=h d=r SL=2 π r h SL - lateral surface

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**l=2πr l = πr S=4 π2rR S=4 π r2 Torus Sphere d=R d = 2r/π R r**

Geometric centroid l = πr d = 2r/π S=4 π r2

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Volume solids Pappus-Guldin’s second theorem states that the volume of a solid of revolution generated by rotating a plane figure F around an external axis is equal to the product of the area A of F and the length of the circumference of radius d equal to the distance between the axis and the geometric centroid (2 π d) V = 2 π d A

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**Cone Cylinder V= π r2h V= (π r2h)/3 A=hr d=r/2 A=(hr)/2 d=r/3**

geometric centroid A=hr d=r/2 V= π r2h Cone r h d geometric centroid A=(hr)/2 d=r/3 V= (π r2h)/3

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**V= 2π2r2R V= 4πr3/3 Torus Sphere A=πr2 d=R A=πr2/2 d=4r/3π R r**

Geometric centroid Torus R A=πr2 d=R r V= 2π2r2R Sphere Geometric centroid A=πr2/2 d=4r/3π r V= 4πr3/3

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**“On the Sphere and Cylinder” Archimedes (225 B.C.)**

The surface area of the sphere is equivalent to the surface area of the cylinder that circumscribes it r 2r Scylinder=2πr∙2r=4 πr2 r Ssphere=4 πr2

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**Archimedes Scylinder=πr2∙2r=2 πr3 Ssphere=4 πr2/3**

The volume of the sphere is equivalent to 2/3 of the cylinder’s volume that circumscribes it r 2r Scylinder=πr2∙2r=2 πr3 r Ssphere=4 πr2/3

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**2πr3 = (4πr3)/3 + (πr3)/3 Archimedes**

The volume of the cylinder having radius r and height 2r is the sum of the volume of the sphere having radius r and that of the cone having base radius r and height 2r r 2r r 2r r 2πr3 = (4πr3)/3 + (πr3)/3

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**Vcylinder = Vbowl + Vhalf_sphere**

Galileo’s bowl h r h Circle (section cone) Annulus (section bowl) Vcone = Vbowl r Vcylinder = Vbowl + Vhalf_sphere Vhalf_sphere = Vcylinder - Vcone

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**Vanti-clepsydra = Vcylinder- 2 Vcone**

Theorem: The sphere volume is equivalent to that of the anti-clepsydra Vanti-clepsydra = Vsphere o Vanti-clepsydra = Vcylinder- 2 Vcone o Vsphere = 2πr3 – (2πr3)/3= (4πr3)/3

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**Special thanks to prof. Cinzia Cetraro for linguistic supervision**

Some of the pictures are taken from Wikipedia

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