maths course exercises
Liceo Scientifico Isaac Newton - Roma
solids of revolution
in accordo con il
Ministero dell’Istruzione, Università, Ricerca
e sulla base delle
Politiche Linguistiche della Commissione Europea
percorso formativo a carattere
tematico-linguistico-didattico-metodologico
scuola secondaria di secondo grado
professor
Tiziana De Santis
solid of revolution
Indice Modulo
Strategies - Before



Prerequisites
Linking to Previous Knowledge and Predicting con questionari basati su
stimoli relativi alle conoscenze pregresse e alle ipotesi riguardanti i contenuti
da affrontare
Italian/English Glossary
Strategies – During



Video con scheda grafica
Keywords riferite al video attraverso esercitazioni mirate
Conceptual Map
Strategies - After

Esercizi:
 Multiple Choice
 Matching
 True or False
 Cloze o Completion
 Flow Chart
 Think and Discuss

Summary per abstract e/o esercizi orali o scritti basati su un questionario
e per esercizi quali traduzione e/o dettato

Web References di approfondimento come input interattivi per test orali e
scritti e per esercitazioni basate sul Problem Solving
Answer Sheets
2
solid of revolution
1
Strategies Before
Prerequisites
Plane
geometry
Geometry in
Geometric
space
transformations
Basic concept of Euclidean
geometry
Straight-lines, planes
and angle in space
Pythagoras’ theorem and
Euclid’s theorem
Surface and volume
Cavalieri’s principle
Areas of polygons
Barycentre of a solid
Barycentre of a curve
Barycentre of a plane figure
Solids of revolution
3
Central symmetry
Axial symmetry
Orthogonal symmetry
solid of revolution
2
Strategies Before
Linking to Previous Knowledge and Predicting

Do you know the conditions of perpendicularity and parallelism between two
straight-lines in the plane?

Do you know the conditions of perpendicularity and parallelism between two
straight-lines in space?

Are you able to calculate the area and the perimeter of a plane figure?

Are you familiar with the concept of central symmetry?

Are you familiar with the concept of axial symmetry?

Are you familiar with the concept of orthogonal symmetry?

Do you know the concept of surface area of a solid?

Do you know the concept of volume of a solid?

Do you know Cavalieri’s principle?

Do you know how to find the barycentre of a triangle, a rectangle and a
circle?
4
solid of revolution
3
Strategies Before
Italian/English Glossary
altezza
height
angolo
angle
anticlessidra
anti-clepsydra
apotema
apothem
asse
axis
baricentro (centroide)
barycentre (geometric centroid)
base
base
cateto
cathetus (pl. catheti)
cerchio (semicerchio)
circle (half-circle)
cilindro (indefinito)
cylinder (infinite)
circonferenza (semicirconferenza)
circumference (half-circumference)
circumscrittibile
circumscribable
cono (indefinito)
cone (infinite)
corona circolare
annulus (pl annuli)
diagonale
diagonal
diametro
diameter
equilatero
equilateral
esterno
external
inscritto
inscribed
lato
side
parallelo
parallel
perpendicolare
perpendicular
piano
plane
quadrato
square
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solid of revolution
raggio
radius (pl. radii)
retta
straight line
rettangolo
rectangle
retto
right
rotazione
revolution (rotation)
scodella
bowl
secante
secant
sfera (semisfera)
sphere (half-sphere)
simmetria
symmetry
solido
solid
Superficie
surface
tangente
tangent
toro
torus
trapezio
trapezium
triangolo
triangle
tronco di cono
truncated cone
vertice
vertex (pl vertices)
volume
volume
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solid of revolution
4
Strategies During
Keywords
1) Circle the solids of revolution:
pyramid – sphere – right prism – torus– octahedron – circumference –
truncated cone – annulus - cylinder – cone
2) Completion:

The volume of a sphere is equivalent to that of the ……………………….

The surface area of a ………..is equivalent to that of the cylinder
circumscribes it.

The …….of the cylinder are obtained from the complete rotation of the
radii of the base.

A cone is called ……….. if its apothem is congruent to the diameter of the
base.

The ……………………………. is also obtained from the rotation of a right
triangle around one of its catheti.
________________________________________________________________
sphere, equilateral, anti-clepsydra, right circular cone, bases
7
solid of revolution
5
Strategies During
Conceptual Map
Complete the conceptual map using the following words:
cone
sphere
n
cylinder
n
anti-clepsydra
n
secant
tangent
n
n
n
S=4 π r2
torus
Revolution
solids
Obtained by rotating
rectangle
right triangle
circle
half-circle
lateral surface
S =π r √ (h2+
r2)
Position of Straigthline/plane
volume
lateral surface
volume
V= π r2h
V=(π r2h)/3
surface
volume
S =2 π r h
V= 4πr3/3
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equivalent
external
solid of revolution
6
Strategies After
Multiple Choice
1) A plane intersects a sphere; the polygon which represents the section is
always
a.
b.
c.
d.
a rectangle
a square
a circle
none of these
2) The cone can be obtained from a complete rotation of
a. a square
b. a triangle
c. a trapezium rectangle
d. none of these
3) A truncated cone can be obtained from a complete rotation of
a.
b.
c.
d.
a square
a triangle
a trapezium rectangle
none of these
4) Indicate which of the following statement is correct:
a. the sphere is equivalent to 1/3 of the cylinder circumscribed
b. the cone inscribed in a cylinder whit base radium 2r and height 2r is
equivalent to Galileo’s bowl
c. the cone is equivalent to the anti-clepsydra
d. none of these
5) The surface area of a sphere measures S=16π, the surface area of the
circumscribed cylinder is:
a.
b.
c.
d.
Greater than 16π
Less than 16π
16π
none of these
9
solid of revolution
6) The surface area of a sphere measures 3π, its volume is:
a.
b.
c.
d.
(π√3)/3
4π/3
(π√3)/2
4√3 π/3
7) The volume of anti-clepsydra is equivalent inscribed in a cylinder equilater of
height 2r is:
a.
b.
c.
d.
π r3
(π r3)/3
2π2r2
4πr3/3
8) Galileo’s bowl is equivalent to:
a.
b.
c.
d.
half-sphere
cone inscribed
cylinder
none of these
10
solid of revolution
7
Strategies After
Matching
Match the words on the right with the correct definition on the left:
1. Cone
a. Solid generated by rotating
the circle
2. Galileo’s bowl
b. Solid generated by rotating
the half-circle
3. Anti-clepsydra
4. Cylinder
c. Complementary double cone
solid circumscribed in a
cylinder
5. Sphere
6. Torus
d. Cylinder minus the inscribed
half-sphere
e. Solid generated by rotating
the rectangle
f. Solid generated by rotating
the triangle rectangle
11
solid of revolution
8
Strategies After
True or False
State if the sentences are true or false.
1. The points that do not belong to the surface of the cylinder and having
distance from the axis smaller than the radius are internal to the surface.
2. The surface area of the cylinder is equivalent to that of the inscribed sphere.
3. A right circular cylinder can be obtained from the rotation of a rectangle
triangle around one of its catheti.
4. The intersection figures of two planes perpendicular to the rotation axis of an
indefinite cylinder are two congruent circles.
5. In a circular right cone the plane of the base is perpendicular to the
generatrix of the cone.
6. The apotheme of the cone is any segment having as extremes the vertex of
the cone and one of the points of the circumference of base.
7. An equilateral cone sectioned by a plane passing through the axis of the
cone is an equilateral triangle.
8. The height of a truncated cone is the distance between the two bases.
9. A plane α is secant a spherical surface S if it has a segment in common with
the surface S.
10. A plane tangent to a sphere S has a circumference in common with the
solid.
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solid of revolution
9
Strategies After
Cloze
Complete the text.
The solids of revolution are generated by the rotation of a…….[1] around a
straight line. In particular:
the cone is generated by the complete rotation of a …. [2] around one of its
catheti, the sphere is generated by the complete rotation of a …[3] around its …
[4], the …. [5] is generated by the complete rotation of a rectangle around one
of its sides.
The surface area of the ….[6] is equivalent to the surface area of the cylinder
that is circumscribed it
The volume of the sphere is equivalent to …. [7] 2/3 of the cylinder’s volume
that is …. [8]
The volume of the cylinder having radius r and height 2r is the sum of the
volume of the …..[9] having radius r and the one of the …. [10] having base
radius r and height 2r
The …. [11] sphere volume is equivalent to that of the anti-clepsydra.
13
solid of revolution
10
Strategies After
Flow Chart
Complete the flow chart referring to the position of a plane in relation to
a spheric surface. You can use the terms listed below: secant- tangent-
external
start
Sphere,
plane
input
plane
false
points in
common
true
false
One
point
true
output
output
end
14
solid of revolution
11
Strategies After
Think and Discuss
The following activity can be performed in a written or oral form. The teacher
will choose the modality, depending on the ability (writing or speaking) that
needs to be developed.
The contexts in which the task will be presented to the students are:
A) the student is writing an article about solids of revolution
B) the student is preparing for an interview on a local TV about solids of
revolution
The student should:
1) Choose one of the following topics:



The parts of the sphere
“On the Sphere and Cylinder” by Archimedes
Theorems of Pappus and Guldin
2) Prepare an article or a debate, outlining the main points of the argument, on
the basis of what has been studied.
3) If the written activity is the modality chosen by the teacher, the student
should provide a written article, indicating the target of readers to whom the
article is addressed and the type of magazine / newspaper / school magazine
where the article would be published.
4)
If the oral activity is the modality chosen by the teacher, the student should
present his point of view on the topics to the whole class and a debate could
start at the end of his presentation.
15
solid of revolution
12
Strategies After
Summary
The solids of revolution are generated by the rotation of a plane figure around a
straight line. In particular:

The cylinder is generated by the complete rotation of a rectangle around
one of its sides;

The cone is generated by the complete rotation of a right triangle around
one of its catheti;

The sphere is generated by the complete rotation of a half-circle around its
diameter;

The torus is generated by rotating a circle around an external coplanar
straight line.
Pappus-Guldin's theorems make it possible to determine the surface area
and volume of solids of revolution. The first theorem states that:
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: S=2 π dl
Thus, for the cylinder and cone the following lateral surfaces are obtained
respectively:
Scone = π r √( h2+ r2)
Scyl = 2 π r h.
For the sphere and torus the total areas are: Storus=4 π2rR
Ssphere=4 π r2.
The 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: V = 2 π d A
The following formulas are thus obtained:
Vcone=(π r2h)/3
Vcyl = π r2h
Vtorus = 2π2r2R
Vsphere = 4πr3/3
Archimedes' "On the Sphere and Cylinder" contains significant results achieved
by the mathematician of Syracuse on the solid rotation, as, for example:

The surface area of the sphere is equivalent to the surface area of the
cylinder that is circumscribed it

The volume of the sphere is equivalent to 2/3 of the cylinder’s volume that
is circumscribed it
16
solid of revolution

The volume of the cylinder having radius r and height 2r is the sum of the
volume of the sphere having radius r and the one of the cone having base
radius r and height 2r
Using Cavalieri's principle it can be shown that:
The volume of Galileo’s bowl is equivalent to the volume of the cone inscribed in
the same cylinder, as the cylinder volume is given by the sum of the total
volume of the half-sphere and the bowl, the volume of the half-sphere is given
by the difference between the volume of the cylinder and the volume of the
cone.
The sphere volume is equivalent to that of the anti-clepsydra.
1) Answer to the following questions. The questions could be answered
in a written or oral form, depending on the teacher’s objectives.
a.
How do you obtain a solid of revolution?
b.
What is the difference between a solid and solid surface?
c.
How do you obtain a right cylinder and right cone
d.
Which symmetries does a sphere have?
e.
What is the relationship between the surface of the cylinder and that of a
sphere?
f.
What is the relationship between the volume of the cylinder and that of a
sphere?
g.
Illustrate the first theorem of Pappus-Guldin.
h.
Illustrate the second theorem of Pappus-Guldin.
2) Write a short abstract of the summary (max 150 words) highlighting
the main points of the video.
17
solid of revolution
Web References
http://www.mathwords.com
An interactive math dictionary with many math words, math terms, math
formulas, pictures, diagrams, tables, and examples
http://mathworld.wolfram.com
Encyclopedia of mathematics
http://www.britannica.com/EBchecked/topic/428841/On-the-Sphereand-Cylinder.
http://en.wikipedia.org/wiki/Archimedes
18
solid of revolution
13
Activities Based on Problem Solving
a) A sphere whose surface is 100 π cm2 is cut by a plane far from the center of
the 3 / 5 of its radius. Determine the relationship between the lateral areas
of the two cones having the circle as common base for the top section and as
vertices the extremes of the diameter perpendicular to the secant plane.
b) The base radius of a right cylinder is 6 cm and height is 9 cm. Determine a
point on the axis V such that the ratio of 4 volumes of two cones having as
bases the bases of the cylinder, and as vertex the point V.
c) In a right circular cylinder the lateral surface is equivalent to 4 / 7 of the
total. Knowing that the total height of the cylinder is 12 cm determine the
volume of the sphere which has radius congruent to half the radius of the
cylinder base.
d) In the Discourses and Mathematical Demonstrations Concerning Two New
Sciences, Galileo Galilei describes the construction of a solid that is
called a bowl considering a hemisphere of radius r and the cylinder
circumscribed on it. The bowl is achieved by removing the hemisphere
from the cylinder. Prove, using
Cavalieri’s principle, that the
bowl has volume equal to the cone inscribed in the cylinder.
(Esame di Stato 2009 liceo scientifico sperimentale - PNI)
e) Prove that the proportion of the total area of an equilateral cylinder to the
surface
of
the
circumscribed
sphere
is
3
to
4.
(Esame di Stato 2004 liceo scientifico sperimentale - PNI)
f)
Demonstrate the equivalence between the volume of the sphere inscribed in
a cylinder and the volume of the anti-clepsydra.
19
solid of revolution
Answer Sheets
Keywords:
1) sphere – torus– truncated cone – cylinder – cone
2) anti-clepsydra, sphere, bases, equilateral, right circular cone
Conceptual Map:
Revolution
solids
Obtained by rotating
rectangle
rigth triangle
cone
cylinder
circle
half-circle
torus
sphere
lateral surface
S =π r √ (h2+
r2)
external
Position of Straigthline/plane
volume
lateral surface
volume
V=(π r2h)/3
tangent
surface
V= π r2h
S=4 π r2
volume
S =2 π r h
V= 4πr3/3
20
secant
equivalent
anti-clepsydra
solid of revolution
Multiple Choice:
1C, 2B, 3C, 4B, 5C, 6C, 7D, 8B
Matching:
1F, 2D, 3C, 4E, 5B, 6A
True or False:
T, T, F, T, F, T, T, T, F, F
Cloze:
[1] plane figure
[2] right triangle [3] half-circle
[6] sphere
[7] 2/3
[8] circumscribed it
[11] sphere
21
[4] diameter
[9] sphere
[5] cylinder
[10] cone
solid of revolution
Flow Chart
start
Sphere,
plane
input
plane
false
points in
common
true
false
external
output
secant
end
22
One
point
true
tangent
solid of revolution
Activities Based on Problem Solving
a.
2
b.
36/5 cm or 9/5 cm
c.
243 π /2 cm3
Materiale sviluppato da eniscuola nell’ambito del protocollo d’intesa con il MIUR
23