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.