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solid of revolution transcript
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Liceo Scientific 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
eniscuola / smartenglish – A cura di Linda Rossi Holden
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Label europeo 2010
solid of revolution transcript
Solids of Revolution
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.
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 cylinder. If
these planes are perpendicular to the rotation axis, then it is called right cylinder.
The cylinder is also obtained from the rotation of a rectangle around one of its sides,
which is called height .
The sides perpendicular to the height are called radii of base.
The bases of the cylinder are obtained by the complete rotation of the radii of the
base.
If we consider a half-line s having “V” as the initial point 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 where α is the acute angle between s and r.
The half-line s describes an infinite conical surface and the point V is called the vertex
of the cone .
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 by 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: a
small cone that is similar to the previous one and a truncated cone.
From the similarity between the two cones of height VH and VH’ respectively , we
have the following theorem.
Theorem: the measure of the areas C and C’, obtained by a parallel section, are in
proportion with the square of their respective heights.
A spheric surface is the boundary formed by the complete rotation of a halfcircumference around its diameter.
The rotation of a half-circle generates a solid, the sphere.
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solid of revolution transcript
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, which is called the symmetry
centre.
Every plane passing through the centre of a sphere is a symmetry plane.
In addition, the straight lines passing through its centre are symmetry axes.
If we call d the distance between the centre of the sphere and a generic straight line
s, and r the radius of the sphere, the straight line can be:

Secant with two points in common A and B.

Tangent with one point in common A.

External with no points in common.
A plane in relation to a sphere may be:

Secant: if the intersection is a circle.

Tangent: if the intersection is a point.

External: if there is no intersection.
The torus is a surface generated by the complete rotation of a circle around an
external axis coplanar with the circle.
The surface area and volume calculus of the rotation solids represent a classic
problem of ancient Greek maths; and to calculate them in this work we will use
Pappus’s centroid theorems.
Pappus of Alexandria was a mathematician who lived between the 3rd-4th century
A.D.; he handed down and reworked some of the works of mathematicians such as
Euclid and Ptolemy.
In his work, entitled “Mathematicae Collectiones”, translated and published in 1589 by
Federico Commandino, he talks about surfaces and volumes.
Guldin was a Swiss mathematician of the 16th century, and in his work entitled
“CENTROBARYCA” he talks about Pappus’s centroid theorems.
In mathematical literature, these theorems are marked with the names of both.
Now, we will present the first theorem relating to the measure of surfaces:
Pappus’s first theorem states that the measure of the area of the surface generated
by the rotation of an arc of curve around an axis, is equal to the product between the
length L of the arc of curve and the measure of the circumference described by its
geometric centroid (2 π d ).
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solid of revolution transcript
S=2πdL
If in a cone l = √(h2+r2) is the segment that generates the surface, and d =r/2 is the
distance between the barycentre and the axis of rotation, the lateral area is given by
this formula S=π r √(h2+r2).
For the cylinder if l = h is the segment that generates the surface, and d =r is the
distance between the barycentre and the axis of rotation, the lateral area is given by
this formula S=2π r h.
As for the torus, the length l of the arc that generates the surface is the measure of
the circumference, having radius equal to r. The barycentre is the centre O of the
circumference and its distance from the axis of rotation is R, so the area of the
surface is this formula S=4 π2rR.
As for the sphere, the length l of the generatrix curve is the half-circumference, d
(d=2r/π ) is the distance between its barycentre and the axis, so the area of the
surface is: S=4 π r2.
Pappus’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=2πdA
As for the cylinder generated by the rotation of a rectangle, having base r and height
h.
A=hr is the area of the rectangle, and d =r/2 is the distance between the
barycentre and the axis , its volume is V=π r2h
The cone generated by the revolution of the triangle of base r and height h. The area
of the triangle is A=hr/2 . The barycentre distance from the axis of rotation is d =r/3
and the volume is V as in this formula V= π r2h/3.
Now, we consider the torus obtained by the revolution of a circle having radius r and
area A as in the formula A= π r2. If d =R is the distance between the centre of the
circle and axis of rotation, its volume is V= 2π2r2R. As for the sphere, if the area of
the half-circle having radius r is A = πr2/2 and if the distance from the barycentre of
the axis of rotation is d =4R/3π the volume is therefore V= 4πr3/3.
Pappus’s theorems are the most brilliant results of antiquity, relating to modern
integral calculus. Pappus’s theorems of centroids can be useful today to calculate the
barycentre of a figure when we know its surface or its volume, through an inverse
process.
The equivalence between the sphere and other solids, was used in ancient times to
calculate its volume.
One of Archimedes' most important works is “On the Sphere and Cylinder”, written in
225 B.C. He achieved essential results on this topic and even wanted a sphere and a
cylinder sculpted on his grave, as reported by Cicero in the “TUSCULANAE
DISPUTATIONES”.
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solid of revolution transcript
He was the first to measure the volume and the surface area of the sphere. He
calculated that the surface area of the sphere is equivalent to the surface area of the
cylinder that circumscribes it.
The volume of the sphere is equivalent to 2/3 of the cylinder’s volume that
circumscribes it.
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.
Let us now consider some applications of Cavalieri’s principle to calculate the volume
of the sphere.
If from the cylinder having radius and height equal to r, we take away the half-sphere
having radius r, we obtain Galileo’s bowl.
And we can calculate the volume of the bowl using Cavalieri’s principle demonstrating
the equivalence with the cone inscribed in the same cylinder which therefore has base
radius r and height r.
In each section with parallel planes the bowl determines an annulus, the area of which
coincides with the section of the cone sectioned at the same height h.
So we can say that the volume of the bowl is equal to the volume of the cone the
volume of the cylinder, is the sum of the volume of the bowl, and the volume of the
half-sphere is the difference between the volume of the cylinder and the volume of
the cone.
If we consider a sphere with centre O an equilateral cylinder circumscribed in it and
two cones with vertices O with bases coinciding with those of the cylinder, we define
as an anti-clepsydra the solid obtained from the difference between the cylinder and
the two cones. It can be demonstrated using Cavalieri’s principle.
The sphere volume is equivalent to that of the anti-clepsydra.
We can thus calculate the volume of the sphere as the difference between the volume
of the cylinder circumscribed and that of the cones complementary to the anticlepsydra.
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