Liceo Scientifico Isaac Newton

Maths course

Solid of revolutionSolid of revolution

Professor

Tiziana De Santis

Read by

Cinzia Cetraro

PP

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

axis

r

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

CylinderThe infinite cylinder is the part of space obtained from the complete rotation of a straight line s around a parallel straight line r

s – generatrix

r – axis

rs

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

rs

base

baseradius

height

The cylinder is also obtained from the rotation of a rectangle around one of its sides

The bases of the cylinder are obtained from the complete rotation of the radii of the base

It is called height

The sides perpendicular to the height are called radii of base

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

r

V

s

α

The half-line s describes an infinite conical surface and

the point V is called vertex of the cone

V

s

α

r

the infinite cone is the part of space obtained from the

complete rotation of the angle α around r

infinite cone infinite conical surface

and a straight line r passing through V called axis

Cone

V

baseH P

VH - height

VP - apothem

HP - radius of base

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

If we section a cone with a plane that is parallel to the

base, we obtain two solids:

H α

V

α’H’

H

H’

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

truncated cone

small cone

truncatedcone

Hp: α // α’

VH ┴ α

Th: C : C’ = VH2 : VH’ 2

Hα

V

α’H’

H

H’

C

C’

Theorem: the measure of the areas C and C’, obtained

from a parallel section, are in proportion with the

square of their respective heigths

Sphere

C C - center P

PC - radius

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 centresymmetry centre Every plane passing through the centre of a sphere is a Every plane passing through the centre of a sphere is a symmetry planesymmetry plane The straight-lines passing through its centre are The straight-lines passing through its centre are symmetry symmetry axesaxes

Positions of a straight line in relation to a spheric surface

C C C

AB

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

EXTERNAL PLANE:

no intersection

TANGENT PLANE:

intersection is a point

SECANT PLANE:

intersection is a circle

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)

Pappus-Guldin’s Centroid Theorem

S = 2 π d l

Surface area calculusSurface 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 π d )

hr

Cylinder

r

h

r/2

l

Cone

l=h d=r

SL=2 π r h

SL=π r √ h2+ r2

l=√h2+r2 d =r/2Geometric centroid

SL - lateral surface

S=4 π2rR

Torus

O

r

R

l=2πr d=R

Sphere

d = 2r/πl = πr

Geometric centroid

S=4 π r2

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

Volume solids

Cylinder

r

hd

geometric centroid

V= (π r2h)/3r

h d

geometric centroid

V= π r2h

A=hr d=r/2

Cone

A=(hr)/2 d=r/3

V= 2π2r2R

R

r

Torus

A=πr2 d=R

Geometric centroid

Sphere

d=4r/3πA=πr2/2

V= 4πr3/3r

Geometric centroid

The surface area of the sphere is equivalent to the

surface area of the cylinder that circumscribes it

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

r

r

2r Scylinder=2πr∙2r=4 πr2

Ssphere=4 πr2

The volume of the sphere is equivalent to 2/3 of the

cylinder’s volume that circumscribes it

r

r

2r 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

2rr

r

2r

= +

Archimedes

2πr3 (4πr3)/3 (πr3)/3

Galileo’s bowl

r

r

rh

Vcone = Vbowl

Vhalf_sphere = Vcylinder - Vcone

Vcylinder = Vbowl + Vhalf_sphere

h

Annulus

(section bowl)

Circle

(section cone)

Theorem: The sphere volume is equivalent to that

of the anti-clepsydra

o

Vsphere = 2πr3 – (2πr3)/3= (4πr3)/3

Vanti-clepsydra = Vsphere

o

Vanti-clepsydra = Vcylinder- 2 Vcone

Some of the pictures are taken from Wikipedia

Special thanks to prof. Cinzia Cetraro

for linguistic supervision