1. Liceo Scientifico Isaac Newton
Maths course
Solid of revolution
Professor
Tiziana De Santis
Read by
Cinzia Cetraro

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. Positions of a straight line in relation to a spheric surfaceB 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

11. 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

12. 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

13. Surface area and volume calculus
Habakkuk Guldin
(1577 –1643)

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. “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

21. 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

22. 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

23. 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

24. 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

25. Special thanks to prof. Cinzia Cetraro for linguistic supervision
Some of the pictures are taken from Wikipedia