11.1 Area & Volume
The student will learn about:
area postulates, Cavalieri’s
Principle, and the areas and
volume of basic shapes.
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1
Perimeter
The idea and formulas for perimeter of all
plane figures is assumed knowledge of the
students.
Ask now if you have any questions.
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§4.6 Area and Volume
Historic idea of area of a rectangle.
Unit square
(2)  (3)
(2.5)  (1.75)
However irrational measures and irregular
shapes were a problem.
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Greek’s Use of Area in Algebra
b
c
a
a (b + c) = ab + ac
distributive law
Completing the square.
Modern Idea of Area
Historic idea of area of a rectangle.
Continue with ever finer grids and use a limit
process.
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Postulates - Hilbert
Area Postulate - To each polygonal region there
corresponds a unique positive real number.
Congruence Postulate - If two triangles are
congruent, then the triangular regions
determined by them have the same area.
Area Addition Postulate - If two polygonal
regions intersect only in edges and vertices (or
do not intersect al all), then the area of their
union is the sum of their areas.
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Postulates - Hilbert
Unit Postulate - The area of a square region is
the square of the length of its side.
Cavalieri’s Principle – next slide please.
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Cavalieri’s Principle for Area
Lx
L’x
If, in two shapes of equal altitude, the sections
made by lines at the same distance from their
respective bases are always equal in length, then
the areas of the shapes are equal.
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Playing with Sketchpad
Area
ABC = 22.82 cm 2
Area P1 = 22.82 cm 2
B
E
D
A
F
J
G
G'
H
K
C
I
H'
I'
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Area of a Parallelogram.
l
w
Remove triangle from one end and slide it to the
other end giving an area of,
K = l x w.
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Area of a Trapezoid.
Average base
Remove two triangle from the bottom and
swivel them to the top giving an area of,
K = ½ ( b 1 + b 2) x h.
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Area of a Triangle.
Remove two triangle from the top and swivel
them to the bottom giving an area of,
K = b x ½ h = ½ bh.
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Area of a Regular Polygon.
a
½p
Rearrange the triangles to form a parallelogram
giving an area of,
K = ½ ap.
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Area of a Circle.
r
½C=πr
Rearrange the triangles to form a parallelogram
giving an area of,
K = π r2 .
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Circular Sector
θ

2
A
r
360
Circular Segment
A = sector - triangle
Postulates - Hilbert
Unit Postulate - The volume of a rectangular
parallelepiped is the product of the altitude and
the area of the base.
Cavalieri’s Principle – next slide please.
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Cavalieri’s Principle for Volmue
If, in two solids of equal altitude, the sections
made by planes parallel to and at the same
distance from their respective bases are always
equal, then the volumes of the two solids are
equal
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Prisms.
A prism is the figure formed when the
corresponding vertices of two congruent
polygons, lying in parallel planes are joined.
The lines joining the corresponding vertices are
called lateral edges. The congruent polygons are
called the bases, and the other surfaces are
called the lateral faces, or as a group, the lateral
surfaces.
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Prisms.
If the lateral edges are perpendicular to the
plane of the bases, the prism is a right prism;
otherwise, it is an oblique prism.
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Volume of a Prism.
Theorem. The volume of any prism is the
product of the altitude and the area of the base.
Use Cavalieri’s Principle and the unit postulate.
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Pyramids.
Given a polygonal region R in a plane E, and a
point V not in E. the pyramid with base R and
vertex V is the union of all segments VQ for
which Q belongs to R.
V
The altitude of the
pyramid is the
perpendicular distance
from V to E.
R
E
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Theorem.
Theorem. If two pyramids have the same
altitude and the same base area, and their bases
lie in the same plane, then they have the same
volume.
Use Cavalieri’s Principle.
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Theorem.
Theorem. The volume of a triangular pyramid
is one-third the product of its altitude and its
base.
In the figure the prism
is divided into three
pyramids of equal
volume.
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Theorem.
Theorem. The volume of any pyramid is onethird the product of its altitude and its base.
Use Cavalieri’s Principle on a triangular pyramid
with area the same base as the given pyramid.
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Truncated Pyramids.
Volume of a truncated pyramid is that of the
full pyramid minus the pyramid cut off the top.
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Cylinders.
Use Cavalieri’s Principle a cylinder may be
treated the same as a prism. Hence the volume
of a cylinder is the product of its altitude and
the area of its base.
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Cones.
Use Cavalieri’s Principle a cone may be treated
the same as a pyramid. Hence the volume of a
cone is one-third the product of its altitude and
the area of its base.
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Sphere
The Problem is to find a shape with a
known volume that compares to a sphere –
What do you thinks works?
Cavalieri’s Method
Hemisphere only!
r 2  h2
r
h
2π(r – h)
VHS = VP =
r
r
h
h
1 2
2
r 2π r = π r 3
3
3
P
h
r
S
2π r
4
Hence the volume of the sphere is  r 3
3
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Sphere
4 3
The volume of the sphere is
r
3
The surface area of the sphere is 4π r 2.
The derivative of the volume!
Platonic Solids
You should be able to find the surface area of
all five of these solids.
You should be able to find the volume of the
first three of these solids.
Summary.
• We learned about area postulates.
• We learned Cavalieri’s Principle for area.
• We learned about the area of basic shapes.
• We learned about volume postulates.
• We learned Cavalieri’s Principle for volume.
• We learned about the volume of basic shapes.
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Assignment: 11.1
Hand out Pick’s Theorem