Download Solid Volume Surface Area Prism V = Bh S.A.= 2B+ Ph Pyramid V

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GEOMETRY
Chapter 12 -­‐ Volume & Surface Area
Solid
12.1
Volume
Surface Area
Prism
V = Bh
S.A. = 2B + Ph
Identifying Parts of Polyhedra
Pyramid
V = 13 Bh12.1 Identifying Parts of Polyhedra
S.A. = B + 12 Pl
A polyhedron is a three-dimensional
solid formed by polygons that enclose
a region.
A polyhedron is a three-dimensional solid formed by polygons that enclose a region.
• A faceforming
of a polyhedron
is one
of the polygons
the solid.
• A face of a polyhedron is
the
solid.
Cylinder
V one
= π of
r 2 hthe polygons
S.A.
= 2π r 2 +forming
2π rh
• A base of a polyhedron is a face on the top and/or bottom of the polyhedron
• A base of a polyhedron is a face on the top(there
and/or
of forthe
may be bottom
up to two bases
eachpolyhedron
position of the polyhedron).
• A lateral face of a polyhedron is any face2 that is not a base.
2
1
Cone (there may be up to twoVbases
of the polyhedron).
= 3 πfor
r heach position
S.A.
= π rformed
+ πbyrlthe intersection of two faces.
• An edge of a polyhedron
is a segment
A lateral
• A lateral face of a polyhedron is any face •that
is edge
notofaa polyhedron
base. is an edge formed by the intersection of two
lateral faces.
Sphere
• A vertex
a polyhedron
is the =
point
intersection
of three or more edges.
• An edge of a polyhedronVis=a43segment
formed
byof the
intersection
π r3
S.A.
4ofπof
r 2two faces.
Examples:
• A lateral edge of a polyhedron is an edge formed by the intersection of two
Quadrilateral ABCD is a face of the polyhedron.
12
lateral faces.
Pentagon CDEFG is a base of the polyhedron.
Quadrilateral ABCD is a lateral face of the polyhedron.
• A vertex of a polyhedron is the point of intersection
of three or more edges.
Segment
is an edge of the polyhedron.
12.1 Iden)fying Parts of PCDolyhedra
Segment BC is a lateral edge of the polyhedron.
Examples:
Point A is a vertex of the polyhedron.
G
E
D
C
B
© 2010 Carnegie Learning, Inc.
12
F
Quadrilateral ABCD is a face of the polyhedron.
Pentagon CDEFG is a base of the polyhedron.
Quadrilateral ABCD is a lateral face of the polyhedron.
Segment CD is an edge of the polyhedron.
Segment BC is a lateral edge of the polyhedron.
Point A is a vertex of the polyhedron.
A
F
12.2
12.2 Calcula)ng Volume and Surface Area of Solids
E
Calculating Volume Gand Surface Area of Solids
The volume of a solid three-dimensional object is the amount of space contained
D
C
inside the object.
The surface area of a solid three-dimensional object is the total area of the outside
surfaces of the solid.
Chapter 12
| Volume and Surface Area
Volume ! 14(7)(7) " 7(4)(7)
B
A
! 686 " 196
! 882
The volume of the object is 882 cubic inches.
Surface area ! 3(14)(7) " 3(7)(4) " 3(7)(7) " 11(7)
! 294 " 84 " 147 " 77
! 602
The surface area of the object is 602 square inches.
12
12.3
7 in.
4 in.
7 in.
11 in.
7 in.
7 in.
14 in.
Calculating Volume and Surface Area of Prisms
A prism is a polyhedron formed by parallelograms connecting the corresponding
sides of two congruent and parallel polygons.
The volume of a prism is the amount of space contained inside the prism.
822
Chapter 12
calculate
the Area
volume of a prism, use the formula V ! Bh, where V is the volume
| To
Volume
and Surface
of the prism, B is the area of a base of the prism, and h is the height of the prism.
© 2010 Carnegie Learning, Inc.
822
Examples:
GEOMETRY
Chapter 12 -­‐ Volume & Surface Area
12.3 Calcula)ng Volume and Surface Area of Prisms
• A right prism is a prism with rectangular faces.
• An oblique prism is a prism that is not a right prism. Faces of an oblique prism are parallelograms.
• The orientaAon of prisms can be different. The base is not always located on the top or boEom.
What effect does doubling the height have on the volume of a prism?
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12.4 Calcula)ng Volume and Surface Area of Pyramids
What effect does doubling the height have on the volume of a pyramid?
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A famous modern-­‐day pyramid sits in front of the Louvre art museum in Paris, France. The pyramid was designed by I. M. Pei, a well-­‐known Chinese-­‐American architect. This square pyramid has a base that has a side length of 115 feet and a height of about 70 feet. Calculate the volume of this pyramid.
________________________
________________________
GEOMETRY
Chapter 12 -­‐ Volume & Surface Area
12.5 Calcula)ng Volume and Surface Area of Cylinders
What effect does doubling the height have on the volume of a cylinder?
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12.6 Calcula)ng Volume and Surface Area of Cones
What effect does doubling the height of a cone have on the volume?
______________________________________________________________________________________
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Archimedes considered his discovery of the relationship between a sphere and
______________________________________________________________________________________
a circumscribed cylinder to be his greatest achievement. In fact, he asked for a
sculpted sphere and cylinder to be placed on his tomb.
In this lesson, you will see how he discovered the formula for the volume of a sphere.
12.7 Calcula)ng Volume and Surface Area of Spheres
Recall, a circle is the set of all points in two dimensions that are equidistant from the
center of the circle. A sphere can be thought of as a three-dimensional circle.
radius
center
12
great circle
diameter
hemisphere
A sphere is the set of all points in three dimensions that are equidistant from a given
point called the center.
Who is Archimedes of Syracuse, Sicily?
The radius of a sphere is a line segment drawn from the center of the sphere to
______________________________________________________________________________________
a point on the sphere.
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The diameter of a sphere is a line segment drawn between two points on the
sphere passing through the center.
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The antipodes of a sphere are the endpoints of the diameter.
through the center of the sphere.
A hemisphere
is half of a sphere bounded by a great circle.
**Assignments: Complete 12.3 -­‐ 12.7**
© 2010 Carnegie Learning, Inc.
A cross section of a three-dimensional figure is the two-dimensional figure formed
Although the Earth is not perfectly spherical, its mean radius is 3, 959 miles. What is the Earth’s surface by the intersection of a plane and a solid when a plane passes through the solid.
area? If the Earth is 71% w
ater, how many square miles of Earth is land?
A great circle of a sphere is a cross section of a sphere when a plane passes
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