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Chapter 12–1 Prisms
A 3–dimensional shape is called a solid. The
polygons that form a solid are called faces. A
solid that has 2 congruent parallel faces is called
a prism. The 2 congruent parallel faces are
called bases. The other polygons are lateral
faces. The height of a prism is the distance
between its bases.
Square prism triangular prism
Lateral Area is the area of all the lateral faces.
Total Area is the lateral area added to the area
of the bases.
Lateral Area
The area of the lateral faces of a right prism is
the perimeter of a base times the height of the
prism to that base.
Total Area
The total area of a right prism is its lateral area
added to the areas of the two bases.
Volume
The volume of a right prism is the area of a base
multiplied by the height of the prism to that
base.
Find the lateral area, total area, and volume of
the prisms on the other page having dimensions
5cm, 5cm, by 6cm & 3cm, 4cm, 5cm, by 6cm.
Chapter 12–2 Pyramids
A solid with a polygonal base and triangular
lateral faces is called a pyramid. The point
where the lateral faces intersect is called the
vertex. The distance from the vertex to the base
is the height. The height of a triangular lateral
face is called a slant height.
square pyramid triangular pyramid
Regular pyramids have the following
properties:
Base is a regular polygon.
Lateral faces are congruent isosceles
triangles.
The lateral faces have the same slant height.
Lateral Area for Regular Pyramids
The area of the lateral faces of a regular
pyramid is half the perimeter of the base times
the slant height.
Total Area
The total area of a regular pyramid is its lateral
area added to the area of the base.
Volume
The volume of a regular pyramid is one third the
area of the base multiplied by the height of the
pyramid.
Find the lateral area, total area, and volume for
the square pyramid 12cm 12cm, h = 8cm.
Chapter 12–3 Cylinders and Cones
A cylinder is like a prism except that its bases
are circles instead of polygons.
h
r
Lateral Area of a Cylinder
The lateral area of a right circular cylinder is the
circumference of a base times the height of the
cylinder.
Total Area of a Cylinder
The total area of a right circular cylinder is
its lateral area added to the areas of the two
bases.
Volume of a Cylinder
The volume of a right circular cylinder is the
area of a base multiplied by the height of the
cylinder.
A cone is like a pyramid except that its base is a
circle instead of a polygon. r2 + h
2 = l
2
h l
r
Lateral Area of a Cone
The lateral area of a right circular cone is half
the circumference of the base times the slant
height.
Total Area of a Cone
The total area of a right circular cone is its
lateral area added to the area of the base.
Volume of a Cone
The volume of a right circular cone is one third
the area of the base multiplied by the height of
the cone.
Chapter 12–4 Spheres
A sphere is the set of all points in space that
are a given distance from a fixed point.
r
Area of a Sphere
The area of a sphere equals 4π times the square
of the radius.
Volume of a Sphere
The volume of a sphere equals
π times the
cube of the radius.
Find the area and volume of the sphere having
radius 5 m.
5 m
The volume of a sphere is 972 cm3. Find its
radius and its area.
Four identical snowballs fit exactly
inside a cylindrical can as shown.
Will the can hold two more
identical snowballs? (r = 10 cm)
Chapter 12–5 Ratios of Areas & Volumes
Ratios of Areas & Volumes:
If the scale factor of two similar solids is a:b,
then
the ratio of the perimeters is a:b
the ratio of the areas is a2:b
2
the ratio of the volumes is a3:b
3
h = 2 h = 4 a:b = 1:2
P = 4 P = 8 a:b = 1:2
LA = 8 LA = 32 a2:b
2= 1:4
TA = 10 TA = 40 a2:b
2= 1:4
V = 2 V = 16 a3:b
3= 1:8
Show that a sphere of radius 2 m and radius 3 m
satisfy the area and volume ratios.
One cylinder has radius 2 in and height 5 in a
second cylinder has radius 6 in and height 15 in.
What is the ratios of their lateral areas, total
areas, and volumes?
Probability Vocabulary
An event is any outcome or set of outcomes
from a probabilistic situation.
A sample space is the set of all possible
outcomes from a probabilistic situation.
Probability is the likelihood that an event will
occur.
When each of the outcomes in the sample space
has an equally likely chance of occurring, the
probability of an event is the ratio of the
number of outcomes in the event to the number
of outcomes in the sample space.
Calculate the probability of rolling an odd
number on a six-sided die numbered 1 to 6.
Event = rolling an odd number = {1, 3, 5}
Sample space = numbers 1 to 6 = {1, 2, 3, 4, 5,
6}
Calculate the probability of picking an ace from
a deck of 52 standard playing cards.
Event = picking an ace = {A , A , A , A }
Sample space = 52 standard playing cards
={ A , 2 , 3 , …, Q , K , A , 2 , 3 , … , Q , K , A , 2
, 3 , … , Q , K , A , 2 , 3 , …, Q , K }
Calculate the probability of tossing a penny,
nickel, and dime and all three land on heads.
HHH means penny lands on heads, nickel lands
on heads, and dime lands on heads
Event = {HHH}
Sample space = {HHH, HHT, HTH, THH, HTT,
THT, TTH, TTT}
Unions, Intersections, & Complements
The complement of an event is all of the
outcomes in the sample space that are not in the
event.
The intersection of two events is the set of
outcomes that are in both the first event and the
second event. The symbol for intersection is .
The union of two events is the set of outcomes
that are in the first event or in the second event
(or in both). The symbol for union is .
Roll a six-sided die numbered 1 to 6. Let event
A be rolling an odd number, let event B be
rolling a four, five, or six.
Sample space = {1, 2, 3, 4, 5, 6}
A = {1, 3, 5} B = {4, 5, 6}
complement of A = {2, 4, 6}
complement of B = {1, 2, 3}
intersection: A and B = A B = {5}
union: A or B = A B = {1, 3, 4, 5, 6}
The probability of the union of two events can
be found by using the Addition Rule:
Pick a card from a standard deck of 52 playing
cards. Event A is picking a club. Event B is
picking a face card (jack, queen, or king). Find
the probability the card is a club or a face card.
A = {A ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10 , J ,
Q , K }
B = {J ,Q ,K ,J ,Q ,K ,J ,Q ,K ,J ,Q ,
K ,}
A or B = A B = {A ,2 ,3 ,4 ,5 ,6 ,7 ,8
,9 , 10 , J , Q , K , J , Q , K , J , Q , K
, J , Q , K }
A and B = A B = {J , Q , K }
Probability Models
When two or more probabilistic situations occur
a probability model can help determine the
sample space and find probability of an event.
Two colored spinners shown below are spun.
Find the sample space and probabilities.
red
blue green
yellow
Probability Area Model
Exactly 2 probabilistic situations
Outcomes of each situation are written along
one side of a rectangle with their
probabilities
The smaller inside rectangles represent the
sample space
The probability of an outcome is the area of
its rectangle.
Area Model
red yellow
blue red, blue yellow, blue
green red, green yellow, green
Tree Diagram
Any number of probabilistic situations
Outcomes of each situation are written at the
end of each branch
The probability of an outcome is written
along its branch
Sample space is at the end of the tree
Tree Diagram
red, blue
blue
½
red red, green
¼ ½ green
blue
¾ ½ yellow, blue
yellow
½ green
yellow, green
Conditional Probability
In a probabilistic situation with two events A
and B but we know that event B has occurred
then the probability of event A given event B
has occurred is the conditional probability
.
For the conditional probability ,
event B has already occurred, so the outcomes
of event B becomes the sample space for
.
The conditional probability is the
fraction of event B’s outcomes that include
event A, called the Multiplication Property:
Roll a six-sided die numbered 1 to 6. Event A
is rolling an odd number. Event B is rolling a
one, two, or three. Find the conditional
probability . “probability of an
odd number given {1, 2, 3}”
A = {1, 3, 5} B = {1, 2, 3}
A and B = {1, 3}
Two events are independent when the outcome
of one does not influence the outcome of the
other.
If A and B are independent events, then
Because knowing event B occurred does not
change the probability of event A occurring.
The multiplication property can be written as
when A and B are independent events.