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Areas and
Volumes of Solids
Chapter 12
Section 12-1
Prisms
POLYHEDRON
is a three-dimensional
figure in which each
surface is a polygon
and
The surfaces are called
faces. Two faces intersect
at an edge, and a vertex is a
point where three or more
edges intersect.
PRISM
Is a polyhedron with two
identical parallel faces.
Each of these faces is
called a base and
a prism is named by the
shape of its bases.
ALTITUDE
• A segment joining the
two base planes and
perpendicular to both
• The length of the
altitude is the height of
the prism
LATERAL FACES
faces that are not
bases
RIGHT PRISM
A prism having
rectangular lateral
faces.
OBLIQUE PRISM
A prism having nonrectangular lateral
faces.
THEOREM 12-1
The lateral area of a
right prism equals the
perimeter of a base
times the height of
the prism
LA = ph
SURFACE AREA OF A
PRISM
Surface Area equals
the sum of the areas of
all its faces
SA = LA +2B
* Also called Total Area
THEOREM 12-2
The volume of a right
prism equals the area
of a base times the
height of the prism.
V = Bh
Section 12-2
Pyramids
PYRAMID
Is a polyhedron with only
one base. The other faces
are triangles that meet at a
vertex and
a pyramid is named by
the shape of its base.
ALTITUDE
Is a segment from the
vertex perpendicular to
the base.
LATERAL FACES
are the triangular
faces.
LATERAL EDGES
The segments where
the lateral faces
intersect.
REGULAR PYRAMID
Base is a regular polygon
All lateral edges are congruent
All lateral faces are congruent
isosceles triangles
The height is called the slant
height
The altitude meets the base at
its center
METHODS FOR FINDING LATERAL
AREA OF A REGULAR PYRAMID
1. Find the area of one
lateral face and multiply
by n where n is the
number of lateral faces
2. Use the formula LA = ½pl
where l is the slant
height
THEOREM 12-4
The volume of a pyramid
equals 1/3 the area of
the base times the
height of pyramid.
V = 1/3Bh
Section 12-3
Cylinders and
Cones
CYLINDER
a three-dimensional
figure having a curved
region with two parallel
congruent circular bases.
Its axis joins the centers
of the two bases.
RIGHT CYLINDER
Is a cylinder where
the segment joining
the center of the
circular bases is an
altitude
ALTITUDE
Is the height of the
cylinder
THEOREM 12-5
The lateral area of a
cylinder equals the
circumference of a base
times the height of the
cylinder.
LA = 2rh
THEOREM 12-6
The volume of a
cylinder equals the
area of a base times
the height of the
cylinder.
2
V = r h
CONE
a three-dimensional
figure having a curved
surface and one circular
base. Its axis is a
segment from the vertex
to the center of the base.
THEOREM 12-7
The lateral area of a
cone equals half the
circumference of the
base times the slant
height
LA = rl
THEOREM 12-8
The volume of a cone
equals one third the
area of the base times
the height of the cone.
2
V = 1/3r h
Section 12-4
Spheres
SPHERE
is the set of points in
space that are the same
distance from a given
point called the center
THEOREM 12-9
The area of a sphere
equals 4 times the
square of the radius
2
A = 4r
THEOREM 12-10
The volume of a
sphere equals 4/3
times the cube of the
radius
3
V = 4/3r
Section 12-5
Areas and Volumes
of Similar Solids
SIMILAR SOLIDS
Solids having the same
shape but not necessarily
the same size
THEOREM 12-11
If the scale factor of two
similar solids is a:b, then
1. The ratio of corresponding
perimeters is a:b
2. The ratio of the base areas,
of the lateral areas, and of
2
2
the total areas is a :b
3. The ratio of the volume is
a3:b3
END
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