Liceo Scientifico Isaac Newton
Maths course
Solid of revolution
Professor
Tiziana De Santis
Read by
Cinzia Cetraro
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
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
s
rr
s – generatrix
r – axis
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
radius
base
Cone
If
we
consider
a anhalf-line
s having
V
The
half-line
s line
describes
infinite conical
surface
and
and
a straight
r passing
through
V called
axis
the point
infinite
cone
is the
partof
ofthe
space
obtained from the
V is
called
vertex
cone
as the initial point
complete rotation of the angle α around r
s
V
V
α
α
r
s
r
infinite cone
infinite conical surface
If the infinite cone is intersected by
The right circular cone is also obtained from the rotation
of a right triangle around one of its catheti
a plane perpendicular to the axis of
A cone is called equilateral if its apothem is congruent to
the diameter of the base
rotation, the portion of the solid
V
bounded between the plane and the
VP - apothem
vertex is called right circular cone
VH - height
base
H
P
HP - radius of base
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
V
H’
H’
small cone
α’
truncated
cone
H
H
α
Theorem: the measure of the areas C and C’, obtained
from a parallel section, are in proportion with the
square of their respective heigths
V
Hp: α // α’
C’ H’ H’
α’
VH ┴ α
Th: C : C’ = VH2 : VH’ 2
H
C
H
α
Sphere
A spheric surface is the boundary formed by the complete
rotation
of a is
half-circumference
around its
diameter
The sphere
completely symmetrical
around
its centre
called
symmetry
The rotation
of a centre
half-circle generate a solid, the sphere
Every plane passing through the centre of a sphere is a
The
centreplane
of the half-circle is the center of the sphere,
symmetry
while
its radius ispassing
the distance
between
all are
points
on the
The straight-lines
through
its centre
symmetry
surface
and the centre
axes
PC - radius
P
C
C - center
Positions of a straight line in relation to a
spheric surface
C
C
C
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
Position of a plane in relation to a
spheric surface
SECANT PLANE:
TANGENT PLANE:
EXTERNAL PLANE:
intersection is a
circle
intersection is a
point
no intersection
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
Surface area and volume
calculus
Habakkuk Guldin
(1577 –1643)
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=2πdl
Cone
Geometric
centroid
l=√h2+r2
d =r/2
r/2
l
h
SL=π r √ h2+ r2
r
Cylinder
r
h
l=h
d=r
SL=2 π r h
SL - lateral surface
Torus
R
r
l=2πr
O
S=4
d=R
π2rR
Sphere
Geometric
centroid
l = πr
S=4
d = 2r/π
π r2
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
πdA
Cylinder
geometric centroid
A=hr
d=r/2
d
h
V= π r2h
r
geometric centroid
Cone
A=(hr)/2
h
d
r
V= (π r2h)/3
d=r/3
Geometric
centroid
R
Torus
A=πr2
r
d=R
V= 2π2r2R
Sphere
Geometric
centroid
A=πr2/2
r
V= 4πr3/3
d=4r/3π
“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
r
Scylinder=2πr∙2r=4 πr2
Ssphere=4 πr2
Archimedes
The volume of the sphere is equivalent to 2/3 of the
cylinder’s volume that circumscribes it
r
2r
r
Scylinder=πr2∙2r=2 πr3
Ssphere=4 πr2/3
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
r
2r
2πr3
2r
r
=
(4πr3)/3
+
(πr3)/3
Galileo’s bowl
h
r
Circle
Annulus
(section bowl)
h
(section cone)
Vcone =r Vbowl
r
Vcylinder = Vbowl + Vhalf_sphere
Vhalf_sphere = Vcylinder - Vcone
Theorem: The sphere volume is equivalent to
that of the anti-clepsydra
Vanti-clepsydra = Vsphere
o
o
Vanti-clepsydra = Vcylinder- 2 Vcone
Vsphere = 2πr3 – (2πr3)/3= (4πr3)/3
Special thanks to prof. Cinzia Cetraro
for linguistic supervision
Some of the pictures are taken from Wikipedia