Verdana 30 pt

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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
s generatrix r axis
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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
and a straight line r passing through V called
axis
The half-line s describes an infinite conical
surface and the point V is called vertex of the
cone
the infinite cone is the part of space obtained
from the complete rotation of the angle a around r
V
a
a
r
s
s
r
infinite conical surface
infinite cone
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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
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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
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Theorem the measure of the areas C and C,
obtained from a parallel section, are in
proportion with the square of their respective
heigths
Hp a // a VH - a 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
C
C
C
B
A
A
Secant d lt r
Tangent d r
External d gt 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
EXTERNAL PLANE no intersection
SECANT PLANE intersection is a circle
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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
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Surface area and volume calculus
Habakkuk Guldin (1577 1643)
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Pappus-Guldins Centroid Theorem
Surface area calculus
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 p d )
S 2 p d l
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Cone
Geometric centroid
Cylinder
lh dr
SL2 p r h
SL - lateral surface
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Torus
R
r
l2pr
dR
O
S4 p2rR
Sphere
Geometric centroid
l pr
d 2r/p
S4 p r2
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Volume solids
Pappus-Guldins 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 p d)
V 2 p d A
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Cylinder
Ahr
dr/2
V p r2h
Cone
A(hr)/2
dr/3
V (p r2h)/3
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Torus
Geometric centroid
R
Apr2
dR
r
V 2p2r2R
Sphere
Geometric centroid
d4r/3p
Apr2/2
V 4pr3/3
r
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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
Scylinder2pr2r4 pr2
Ssphere4 pr2
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Archimedes
The volume of the sphere is equivalent to 2/3 of
the cylinders volume that circumscribes it
Scylinderpr22r2 pr3
Ssphere4 pr2/3
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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


(4pr3)/3
(pr3)/3
2pr3
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Galileos bowl
Annulus (section bowl)
Vcone Vbowl
Vcylinder Vbowl Vhalf_sphere
Vhalf_sphere Vcylinder - Vcone
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Theorem The sphere volume is equivalent to that
of the anti-clepsydra
Vanti-clepsydra Vsphere
Vanti-clepsydra Vcylinder- 2 Vcone
o
Vsphere 2pr3 (2pr3)/3 (4pr3)/3
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Special thanks to prof. Cinzia Cetraro for
linguistic supervision
Some of the pictures are taken from Wikipedia