Transcript 11.5 Volumes of Prisms and Cylinders

Geometry
11.5 Surface Area & Volume of
Prisms & Cylinders
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11.5 Essential Question
How do you find the surface area and
volume of a prism or cylinder?
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Geometry 12.2 Surface Area of Prisms and Cylinders
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Goals
Know what a prism is and be able to find the
surface area and volume.
 Know what a cylinder is and be able to find the
surface area and volume.
 Solve problems using prisms and cylinders.

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Geometry 12.2 Surface Area of Prisms and Cylinders
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Prism
A polyhedron with two congruent faces, called
the bases.
 The bases are parallel.
 The segments forming the bases are base
edges.
 The other faces are parallelograms and are
called lateral faces.
 The segments joining corresponding vertices of
the bases are lateral edges.

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Parts of a Prism
Base
Lateral
Edges
Lateral
Face
Lateral
Face
Base
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Prism
Right Prism - all lateral faces are rectangles.
 Oblique Prism - has at least one non-rectangular
lateral face.

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How do you find the surface area of
a right prism?
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Surface Area

The surface area of a right prism can be
found using
SA = 2B + PH
B is the area of each base
 P is the perimeter of a base
 H is the height

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Geometry 12.2 Surface Area of Prisms and Cylinders
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Volume
The number of cubic units contained in a
solid.
 Measured in cubic units.

1
1
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V = 1 cu. unit
1
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Prism: V = Bh

B = area of the base, h = height
B
B
h
h
B
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Geometry 12.4 Volume of Prisms and Cylinders
h
10
Cavalieri’s Principle
If the area of cross sections and heights of two
solids are equal, then the volumes are equal.
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Geometry 12.4 Volume of Prisms and Cylinders
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Right Prism
SA = 2B + PH
V= BH
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SA = Surface Area
P = Perimeter of the Base
H = Height of the Prism
B = Base Area (Area of the Base)
V = volume
Geometry 12.2 Surface Area of Prisms and Cylinders
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Example1:
Name the solid and then find its surface area and volume.
4
6
25
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Geometry 12.2 Surface Area of Prisms and Cylinders
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Example 2:
Name the solid and then find its surface area and volume.
6
6
6
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4
Geometry 12.2 Surface Area of Prisms and Cylinders
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Example 3:
Name the solid and then find its surface area and volume.
Triangular Prism
8
3
10
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Geometry 12.4 Volume of Prisms and Cylinders
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Example 4:
Name the solid and then find its surface area and volume.
12
10
12
12
? 6? 3
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Geometry 12.2 Surface Area of Prisms and Cylinders
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Example 5:
Name the solid and then find its surface area and volume.
5
11
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Example 6
4
L
5
A metal bar has a volume of 2400 cm3. The sides
of the base measure 4 cm by 5 cm. Determine the
length of the bar.
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Example 6 Solution
4
5
L
 Method
V
1
 Method
= Bh
 B = 4  5 = 20
 2400 = 20h
 h = 120 cm
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V = L W  H
 2400 = L  4  5
 2400 = 20L
 L = 120 cm
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Cylinder
A prism with congruent circular bases.
 May be right or oblique, just like prisms.

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Surface Area of a Cylinder
Take a cylinder
and cut it apart…
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You get two circles
and a rectangular
area.
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Surface Area of a Cylinder
r
2rh
h
r2
Area of the rectangle.
2r
circumference of the circle.
r2
area of one circle
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Surface Area of a Cylinder
r
2rh
h
2r
The surface area of the cylinder
is:
r2
r2
SA = 2r2 + 2rh
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Geometry 12.2 Surface Area of Prisms and Cylinders
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Surface Area of a Cylinder
r
𝑆𝐴 =
2
2𝜋𝑟
+ 2𝜋𝑟ℎ
h
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Cylinder: V =
2
r h
r
B
h
h
V = Bh
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Cylinder:
SA = 2𝜋𝑟 2 + 2𝜋𝑟𝐻
r = radius
H = Height of the Solid
SA = Surface Area
V = Volume
2
V = 𝜋𝑟 𝐻
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Example 7:
Name the solid and then find its surface area and volume.
12
10
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Example 8:
Name the solid and then find its surface area and volume.
d = 2 in.
r = 1 in.
14 in.
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Example 9: Find the height.
4
h
SA = 96𝝅
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Example 10
Find the diameter of the can.
3 in
V = 115 in3
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Summary
A prism is a polyhedron with 2 congruent
bases and parallelogram lateral faces.
 A cylinder has 2 congruent circular bases,
but it is not a polyhedron.
 Prisms & cylinders may be right or oblique.
 The volumes of prisms and cylinders are
essentially the same:

SA = 2B + Ph
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𝑆𝐴 =
2
2𝜋𝑟
Geometry 12.2 Surface Area of Prisms and Cylinders
+ 2𝜋𝑟ℎ
31
Summary

The volumes of prisms and cylinders are
essentially the same:
V = Bh
&
V=
2
r h
where B is the area of the base, h is the
height of the prism or cylinder.
 Use what you already know about area of
polygons and circles for B.

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Problem 1:
48
44
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A manufacturer of concrete
sewer pipe makes a pipe
segment that has an
outside diameter (o.d.) of
48 inches, an inside
diameter (i.d.) of 44 inches,
and a length of 52 inches.
Determine the volume of
concrete needed to make
52 one pipe segment.
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Problem 1 Solution
Strategy:
Find the area of the ring at
the top, which is the area
of the base, B, and multiply
by the height.
View of the Base
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Problem 1 Solution
48
44
Strategy:
Find the area of the ring at
the top, which is the area
of the base, B, and multiply
by the height.
Area of Outer Circle:
Aout = (242) = 576
52 Area of Inner Circle:
Ain = (222) = 484
Area of Base (Ring):
ABase = 576 - 484 = 92
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V = Bh
Problem 1 Solution
48
ABase = B = 92
44
V = (92)(52)
V = 4784
52
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V  15,029.4 in3
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This one!
Problem 2: Which Holds More?
2.3 in
4 in
3.2 in
4.5 in
1.6 in
2
V  (3.2)(1.6)(4)
 20.48
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 2.3 
V  
  4.5 
 2 
 18.7
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Problem 3: What would the height
of cylinder 2 have to be to have the
same volume as cylinder 1?
r=3
r=4
#1
8
#2
h
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Solution
r=4
#1
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 
V  4 8
2
 128
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Solution
 
r=3
128   3 h
2
128
h
9
h  14.2
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Geometry 12.4 Volume of Prisms and Cylinders
#2
h
40
Homework
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