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In this article, we will discuss various measuring techniques as well as the quantities that can be measured. We will discuss measure of lengths, areas, and volumes of different entities.

1.
The word geometry comes from Greek,

and it means “to measure the earth”.

As a matter of fact, we cannot accurately

and efficiently work with geometrical

objects without measurement.

and it means “to measure the earth”.

As a matter of fact, we cannot accurately

and efficiently work with geometrical

objects without measurement.

2.
Section 13.1

Measurement with Nonstandard and

Standard units

Measurement with Nonstandard and

Standard units

3.
Today’s world is more mathematical

than yesterday’s, and tomorrow’s

world will be more mathematical than

today’s. As computers increase in

power, some parts of mathematics

become less important while others

become more important. While

arithmetic proficiency may have been

“good enough” for many in the

middle of the century, anyone whose

mathematical skills are limited to

computation has little to offer today’s

society that is not done better by just

an inexpensive machine.

From Everybody Counts

than yesterday’s, and tomorrow’s

world will be more mathematical than

today’s. As computers increase in

power, some parts of mathematics

become less important while others

become more important. While

arithmetic proficiency may have been

“good enough” for many in the

middle of the century, anyone whose

mathematical skills are limited to

computation has little to offer today’s

society that is not done better by just

an inexpensive machine.

From Everybody Counts

4.
Measurements

Loosely speaking, to measure a physical quantity is to given a number to

the quantity.

More precisely, to measure a physical quantity is to estimate the ratio of

the magnitude of that given quantity to the magnitude of a unit of the

same type.

Counting is the simplest form

of measurement. When we

count, we are trying to find a

number that can express the

quantity of something, such as

apples in a basket.

The object – apple in this case,

that we count will be used as a

standard for comparison, and

is called a unit.

Loosely speaking, to measure a physical quantity is to given a number to

the quantity.

More precisely, to measure a physical quantity is to estimate the ratio of

the magnitude of that given quantity to the magnitude of a unit of the

same type.

Counting is the simplest form

of measurement. When we

count, we are trying to find a

number that can express the

quantity of something, such as

apples in a basket.

The object – apple in this case,

that we count will be used as a

standard for comparison, and

is called a unit.

5.
However, counting is not

always a precise form of

measurement.

For example, the apples in this

case are not all of the same

size. Hence knowing the

number of apples does not

implying knowing the total

volume or weight of the

apples.

always a precise form of

measurement.

For example, the apples in this

case are not all of the same

size. Hence knowing the

number of apples does not

implying knowing the total

volume or weight of the

apples.

6.
Moreover, not all physical quantities are discrete or countable. If we

examine those quantities such as length, weight, and volume, we

seldomly can find any natural units to use (except possibly the length of

a day), and consequently artificial units have to be created arbitrarily.

Historically, many units of measure were parts of the body, such as

the span, the foot, the yard, the mouthful etc. They were chosen

mainly because of their availability and hence familiarity.

These are called non-standard units because they are not precise and

can vary a great deal from person to person.

examine those quantities such as length, weight, and volume, we

seldomly can find any natural units to use (except possibly the length of

a day), and consequently artificial units have to be created arbitrarily.

Historically, many units of measure were parts of the body, such as

the span, the foot, the yard, the mouthful etc. They were chosen

mainly because of their availability and hence familiarity.

These are called non-standard units because they are not precise and

can vary a great deal from person to person.

7.
Making Measurements

with non-standard units

Reflection from Research

When learning about measurement, children should use informal, non-

standard units before being introduced to standard units and measuring

tools. (Van de Walle, 1994)

The Curriculum and Evaluation Standards for School Mathematics

supports the use of non-standard units in the classroom because it can

help them to

(1) understand the process of making measure and the purpose of units.

(2) recognize the necessity of standard units in order to communicate.

with non-standard units

Reflection from Research

When learning about measurement, children should use informal, non-

standard units before being introduced to standard units and measuring

tools. (Van de Walle, 1994)

The Curriculum and Evaluation Standards for School Mathematics

supports the use of non-standard units in the classroom because it can

help them to

(1) understand the process of making measure and the purpose of units.

(2) recognize the necessity of standard units in order to communicate.

8.
Children may have difficulties in learning measurements due to the lack

of life experience. They may not have any need to compare the lengths of

two objects that cannot be placed side by side, to communicate the

weight of an object accurately over the phone, or to do something that

requires precision.

“Teachers should guide students’ experience by making the resources

for measuring available, planning opportunities to measure, and

encouraging students to explain the results of their actions. … ’’

Principles and Standards for School

of life experience. They may not have any need to compare the lengths of

two objects that cannot be placed side by side, to communicate the

weight of an object accurately over the phone, or to do something that

requires precision.

“Teachers should guide students’ experience by making the resources

for measuring available, planning opportunities to measure, and

encouraging students to explain the results of their actions. … ’’

Principles and Standards for School

9.
News Clip

My wife and I stopped for lunch in

a Nebraska town on our way to

California, and I asked the waitress

how much snow the area usually

got. “About as deep as a meter,” she

replied.

Impressed by her use of the metric system, I asked where she had

learned it. She was momentarily baffled, then she said, “That’s

the one I mean,” pointing out the window to the parking meter in

front of the restaurant.

N. A.

My wife and I stopped for lunch in

a Nebraska town on our way to

California, and I asked the waitress

how much snow the area usually

got. “About as deep as a meter,” she

replied.

Impressed by her use of the metric system, I asked where she had

learned it. She was momentarily baffled, then she said, “That’s

the one I mean,” pointing out the window to the parking meter in

front of the restaurant.

N. A.

10.
Congruence

Two objects are said to be congruent if they have the same shape and

same size.

In particular, we can ignore their colors, textures, and weight etc.

For objects in 2D, two of them are congruent if one can be placed on

top of the other in such a way that their boundaries match up

Two objects are said to be congruent if they have the same shape and

same size.

In particular, we can ignore their colors, textures, and weight etc.

For objects in 2D, two of them are congruent if one can be placed on

top of the other in such a way that their boundaries match up

11.
Measurement of Length

When we compare the length of an object (e.g. a pencil), we ignore all

other physical attributes of that object, such as its width and weight.

We can use a line segment (which has no width) to represent the length

of the object, and hence

Procedure for measuring the length of a line segment

(1) Choose a unit segment

(2) Use a minimum number of congruent copies of the unit to cover

up the line segment while making sure that the units are end to

end, and lined up with the line segment.

(3) Count the number of units used in step (2).

It is highly unlikely that the length of the given segment will be

exactly a whole number multiple of the unit, and this will be one

of the perfect reasons to learn and use fractions and decimals.

When we compare the length of an object (e.g. a pencil), we ignore all

other physical attributes of that object, such as its width and weight.

We can use a line segment (which has no width) to represent the length

of the object, and hence

Procedure for measuring the length of a line segment

(1) Choose a unit segment

(2) Use a minimum number of congruent copies of the unit to cover

up the line segment while making sure that the units are end to

end, and lined up with the line segment.

(3) Count the number of units used in step (2).

It is highly unlikely that the length of the given segment will be

exactly a whole number multiple of the unit, and this will be one

of the perfect reasons to learn and use fractions and decimals.

12.
Measure the length of a pencil with paper clips

The paperclips must be aligned with the

pencil, and there should be no gap

between them and they cannot overlap

each other.

It is highly unlikely that the

length of the pencil will be

exactly a whole number

multiple of the unit, and this will

be one of the perfect reasons to

learn and use fractions and

decimals.

The paperclips must be aligned with the

pencil, and there should be no gap

between them and they cannot overlap

each other.

It is highly unlikely that the

length of the pencil will be

exactly a whole number

multiple of the unit, and this will

be one of the perfect reasons to

learn and use fractions and

decimals.

13.
Measuring the length of a curve with paper clips

Note: the paperclips must be placed to follow the

curve as closely as possible.

Note: the paperclips must be placed to follow the

curve as closely as possible.

14.
Measurement of Area

Procedure for measuring the area of a region

(1) Choose a unit

(2) Use a minimum number of congruent copies of the unit to cover

up the region while making sure that there is no gap and no

overlap between the units.

(3) Count the number of units used in step (2).

What shapes are convenient for using as a unit?

Any shape that Tessellates the plane.

We have seen that squares are good, but are there any other possibilities?

Can we use congruent circles as units?

No, because they either leave gaps in between or have to overlap.

Procedure for measuring the area of a region

(1) Choose a unit

(2) Use a minimum number of congruent copies of the unit to cover

up the region while making sure that there is no gap and no

overlap between the units.

(3) Count the number of units used in step (2).

What shapes are convenient for using as a unit?

Any shape that Tessellates the plane.

We have seen that squares are good, but are there any other possibilities?

Can we use congruent circles as units?

No, because they either leave gaps in between or have to overlap.

15.
How to tile a rectangular floor?

This is our unit.

This is our unit.

16.
How to measure the area of an irregular region?

1) Put a grid on top of the region.

2) Count the number of squares completely inside the region. Ans: 38

3) Count the number of squares partially inside the region. Ans: 18

4) Add half of the result in step (3) to that in step (2). This will be the

approx. area of the region.

Ans: 38 + 18/2 = 47

1) Put a grid on top of the region.

2) Count the number of squares completely inside the region. Ans: 38

3) Count the number of squares partially inside the region. Ans: 18

4) Add half of the result in step (3) to that in step (2). This will be the

approx. area of the region.

Ans: 38 + 18/2 = 47

17.
We can see that the area of a region with curve boundaries cannot

be measured precisely, how can we make the approximation a little

bit better?

Each small square

is ¼ of a large unit

Answer: Use smaller units, or cut each original unit into 4 congruent parts.

Number of units completely inside: 37 × 4 + 17 = 165

Number of units on boundary: 39

Approx. area = 165 + (39) ÷2 = 184.5 small units (= 46.125 large units)

be measured precisely, how can we make the approximation a little

bit better?

Each small square

is ¼ of a large unit

Answer: Use smaller units, or cut each original unit into 4 congruent parts.

Number of units completely inside: 37 × 4 + 17 = 165

Number of units on boundary: 39

Approx. area = 165 + (39) ÷2 = 184.5 small units (= 46.125 large units)

18.
Measurement of Volume

Procedure for measuring the volume of a solid object

(1) Choose a unit

(2) Use a minimum number of congruent copies of the unit to make

a copy of the solid object while making sure that there is no gap

between the units.

(3) Count the number of units used in step (2).

What shapes are convenient for using as a unit?

We have seen that cubes are good, but are there any other possibilities?

Procedure for measuring the volume of a solid object

(1) Choose a unit

(2) Use a minimum number of congruent copies of the unit to make

a copy of the solid object while making sure that there is no gap

between the units.

(3) Count the number of units used in step (2).

What shapes are convenient for using as a unit?

We have seen that cubes are good, but are there any other possibilities?

19.
We can easily see that this method works well for rectangular objects.

How can we measure the volume of round objects?

Displacement Method (for measuring

is a technique used to measure the volume of

irregularly shaped objects.

The object to be measured is placed into a

large body of liquid (usually water), and the

amount of liquid being displaced (or pushed

aside) is then measured.

The volume of the liquid displace will be

exactly the same as the volume of the object

because liquid is not compressible.

How can we measure the volume of round objects?

Displacement Method (for measuring

is a technique used to measure the volume of

irregularly shaped objects.

The object to be measured is placed into a

large body of liquid (usually water), and the

amount of liquid being displaced (or pushed

aside) is then measured.

The volume of the liquid displace will be

exactly the same as the volume of the object

because liquid is not compressible.

20.
Displacement Method

Procedure for measuring the volume of an irregular solid object

(1) Choose a unit

(2) Half fill a sufficiently large container with water (or any other

suitable liquid)

(3) Submerge the given object completely in water

(4) Mark the new water level

(5) Remove the object (without removing any water)

(6) Repeat adding congruent copies of the unit to the container until

the water level matches the mark.

(7) Count the number of units used in step (6).

(1) Do we have to use cubes as units? Or can we use just any shape?

(2) What if the solid absorbs water, such as a donut?

Procedure for measuring the volume of an irregular solid object

(1) Choose a unit

(2) Half fill a sufficiently large container with water (or any other

suitable liquid)

(3) Submerge the given object completely in water

(4) Mark the new water level

(5) Remove the object (without removing any water)

(6) Repeat adding congruent copies of the unit to the container until

the water level matches the mark.

(7) Count the number of units used in step (6).

(1) Do we have to use cubes as units? Or can we use just any shape?

(2) What if the solid absorbs water, such as a donut?

21.
Why does the displacement Method Work?

Dissection properties of Volume

If a solid object is divided into two or more pieces, and the pieces are

rearranged to form another solid object, then the volume of the new

object is the same as the original object.

The Displacement Method is an extreme application of the Dissection

property, where an object is divided into a huge number of pieces,

and then pieces are rearranged into a simple shape, such as a

rectangular block.

Dissection properties of Volume

If a solid object is divided into two or more pieces, and the pieces are

rearranged to form another solid object, then the volume of the new

object is the same as the original object.

The Displacement Method is an extreme application of the Dissection

property, where an object is divided into a huge number of pieces,

and then pieces are rearranged into a simple shape, such as a

rectangular block.

22.
Common Standard Units

We will study both the American and the Metric

systems for the following physical quantities.

Length,

Weight, and

(fluid) Volume

We will study both the American and the Metric

systems for the following physical quantities.

Length,

Weight, and

(fluid) Volume

23.
DID YOU KNOW

It’s a metric world

The united states is the only

western country not presently

using the metric system as its

primary system of measurement.

The only other countries in the

world not using metric system as

their primary system of

measurement are Yemen, Brunei,

and a few small islands; see

Fig. 8.15.

It’s a metric world

The united states is the only

western country not presently

using the metric system as its

primary system of measurement.

The only other countries in the

world not using metric system as

their primary system of

measurement are Yemen, Brunei,

and a few small islands; see

Fig. 8.15.

24.
DID YOU KNOW

In 1906, there was a major effort to convert to the

metric system in the United States, but it was

opposed by big business and the attempt failed.

The Trade Act of 1988 and other legislation declare

the metric system the preferred system of weights

and measures of the U.S. trade and commerce, call

for the federal government to adopt metric

specifications, and mandate the Commerce

Department to oversee the program. The conversion

is currently under way; however, the metric system

has not become the system of choice for most

Americans’ daily use.

In 1906, there was a major effort to convert to the

metric system in the United States, but it was

opposed by big business and the attempt failed.

The Trade Act of 1988 and other legislation declare

the metric system the preferred system of weights

and measures of the U.S. trade and commerce, call

for the federal government to adopt metric

specifications, and mandate the Commerce

Department to oversee the program. The conversion

is currently under way; however, the metric system

has not become the system of choice for most

Americans’ daily use.

25.
DID YOU KNOW

Lost in space

In September 1999, the United

States lost the Mars Climate

Orbiter as it approached Mars. The

loss of the $125 million spacecraft

was due to scientists confusing

English units and metric units.

Two spacecraft teams, one at

NASA’s Jet Propulsion Lab (JPL)

in Pasadena, CA, and the other at a

Lockheed Martin facility in

Colorado, where the spacecraft

was built, were unknowingly

exchanging some vital information

in different units. The missing Mars Climate Orbiter

Lost in space

In September 1999, the United

States lost the Mars Climate

Orbiter as it approached Mars. The

loss of the $125 million spacecraft

was due to scientists confusing

English units and metric units.

Two spacecraft teams, one at

NASA’s Jet Propulsion Lab (JPL)

in Pasadena, CA, and the other at a

Lockheed Martin facility in

Colorado, where the spacecraft

was built, were unknowingly

exchanging some vital information

in different units. The missing Mars Climate Orbiter

26.
DID YOU KNOW

Lost in space

The spacecraft team in Colorado used

English units of pounds of force to

describe small forces needed to adjust

the spacecraft’s orbit. The data was

shipped via computer, without units,

to the JPL, where the navigation team

was expecting the to receive the data

in metric measure.

The mix-up in units led to the JPL

scientists giving the spacecraft’s

computer wrong information, which

threw the spacecraft off course. This

in turn led to the spacecraft entering

the Martian Atmosphere, where it

burned up. The missing Mars Climate Orbiter

Lost in space

The spacecraft team in Colorado used

English units of pounds of force to

describe small forces needed to adjust

the spacecraft’s orbit. The data was

shipped via computer, without units,

to the JPL, where the navigation team

was expecting the to receive the data

in metric measure.

The mix-up in units led to the JPL

scientists giving the spacecraft’s

computer wrong information, which

threw the spacecraft off course. This

in turn led to the spacecraft entering

the Martian Atmosphere, where it

burned up. The missing Mars Climate Orbiter

27.
DID YOU KNOW

Lost in space

On Jan. 3, 1999, NASA launched the $165

million Mars Polar Lander. All radio contact

was lost Dec. 3 as the spacecraft approached

the red planet.

A NASA team that investigated the loss of the Mars Polar

Lander concluded a rocket engine shut off prematurely (due

to programming error) during landing, leaving the

spacecraft to plummet about 130 feet to almost certain

destruction on the Martian surface.

Lost in space

On Jan. 3, 1999, NASA launched the $165

million Mars Polar Lander. All radio contact

was lost Dec. 3 as the spacecraft approached

the red planet.

A NASA team that investigated the loss of the Mars Polar

Lander concluded a rocket engine shut off prematurely (due

to programming error) during landing, leaving the

spacecraft to plummet about 130 feet to almost certain

destruction on the Martian surface.

28.
Length

American standard Metric standard

1 mile = 1760 yards 1 kilometer = 1000 meters

= (5280 feet) 1 meter = 10 decimeters

1 yard = 3 feet 1 decimeter = 10 centimeters

1 foot = 12 inches ( 1 meter = 100 centimeters)

1 mil = 1/1,000 inch 1 centimeter = 10 millimeters

Plastic bags

Conversion:

1 inch is defined to be exactly 2.54 cm in July, 1959.

(before this, the UK inch measures 2.53998 cm, while the US

inch was 2.540005 cm)

American standard Metric standard

1 mile = 1760 yards 1 kilometer = 1000 meters

= (5280 feet) 1 meter = 10 decimeters

1 yard = 3 feet 1 decimeter = 10 centimeters

1 foot = 12 inches ( 1 meter = 100 centimeters)

1 mil = 1/1,000 inch 1 centimeter = 10 millimeters

Plastic bags

Conversion:

1 inch is defined to be exactly 2.54 cm in July, 1959.

(before this, the UK inch measures 2.53998 cm, while the US

inch was 2.540005 cm)

29.
Do you know?

The word “mile” comes from the Latin mille passum,

literally “thousand paces,” a unit introduced to Britain by

the Roman occupation (57 BCE–450 CE).

Each Roman pace is equal to 5 Roman feet, hence

there were 5000 Roman feet in a Roman mile.

But why does the British mile have grown to 5280 feet?

The British has another unit “furlong” for measuring length

and area. A furlong has 660 ft and it does not divide into

5000 evenly. So Elisabeth I defined a mile to be 8 furlongs

and hence 5280 feet.

A furlong is the distance a team of oxen could plow without

needing a rest.

The word “mile” comes from the Latin mille passum,

literally “thousand paces,” a unit introduced to Britain by

the Roman occupation (57 BCE–450 CE).

Each Roman pace is equal to 5 Roman feet, hence

there were 5000 Roman feet in a Roman mile.

But why does the British mile have grown to 5280 feet?

The British has another unit “furlong” for measuring length

and area. A furlong has 660 ft and it does not divide into

5000 evenly. So Elisabeth I defined a mile to be 8 furlongs

and hence 5280 feet.

A furlong is the distance a team of oxen could plow without

needing a rest.

30.
Historical Note

The kilometer was first

defined by the French

Academy of Science in 1791

as the romantic one ten-

thousandth of the length of

the meridian through Paris

from the North pole to the

The kilometer was first

defined by the French

Academy of Science in 1791

as the romantic one ten-

thousandth of the length of

the meridian through Paris

from the North pole to the

31.
Weight

American Standard Metric Standard

1 ton = 2000 pounds 1 (metric) tonne = 1000 kilograms

1 pound = 16 ounces 1 kilogram = 1000 grams

1 gram = 1000 milligrams

Conversion:

1 pound = 0.453 592 37 kilograms

hence

1 kilogram 2.2 pounds

American Standard Metric Standard

1 ton = 2000 pounds 1 (metric) tonne = 1000 kilograms

1 pound = 16 ounces 1 kilogram = 1000 grams

1 gram = 1000 milligrams

Conversion:

1 pound = 0.453 592 37 kilograms

hence

1 kilogram 2.2 pounds

32.
The Gimli Glider - a mixed up in units

On July 23, 1983 Air Canada Flight

143 (a brand new Boeing 767) ran

out of fuel while en routing to

Edmonton from Montreal at 26,000

feet.

Miraculously the caption was able

to land the plane on an abandoned

Royal Canadian Air Force Base at

Gimli, where the runways were

converted into two lane dragstrips

for auto racing.

No one was killed.

On July 23, 1983 Air Canada Flight

143 (a brand new Boeing 767) ran

out of fuel while en routing to

Edmonton from Montreal at 26,000

feet.

Miraculously the caption was able

to land the plane on an abandoned

Royal Canadian Air Force Base at

Gimli, where the runways were

converted into two lane dragstrips

for auto racing.

No one was killed.

33.
The Gimli Glider - a mixed up in units

This mistake was caused by the

ignorance of metric units. The new

767 uses liters and kg to compute

fuel consumption while the crew and

refuelers were only familiar with

pounds and gallons.

They used 1.77lb/liter instead of

The fuel quantity information

system was inoperative before

the flight was started in

Video clip

This mistake was caused by the

ignorance of metric units. The new

767 uses liters and kg to compute

fuel consumption while the crew and

refuelers were only familiar with

pounds and gallons.

They used 1.77lb/liter instead of

The fuel quantity information

system was inoperative before

the flight was started in

Video clip

34.
(Go to the next page)

35.
Click to play

36.
by the way …

The abbreviation for the

pound, lb, comes from

the Latin libra, meaning

A dollar bill weighs about 1 gram,

a dime weighs about 2 grams,

and a quarter 5 grams.

The abbreviation for the

pound, lb, comes from

the Latin libra, meaning

A dollar bill weighs about 1 gram,

a dime weighs about 2 grams,

and a quarter 5 grams.

37.
by the way …

is a unit of mass used for diamonds and other precious stones.

The word carat comes from the Greek keration, a carob bean;

the seed of a Mediterranean evergreen tree.

Traditionally the carat was equal

to 4 grains. The definition of the

grain differed from one country

to another, but typically it was

about 50 milligrams and thus the

carat was about 200 milligrams.

In the U. S. and Britain, the diamond carat was formerly defined

by law to be 3.2 troy grains, which is about 207 milligrams.

Jewelers everywhere now use a metric carat defined in 1907 to

be exactly 200 milligrams or 0.2 gram.

is a unit of mass used for diamonds and other precious stones.

The word carat comes from the Greek keration, a carob bean;

the seed of a Mediterranean evergreen tree.

Traditionally the carat was equal

to 4 grains. The definition of the

grain differed from one country

to another, but typically it was

about 50 milligrams and thus the

carat was about 200 milligrams.

In the U. S. and Britain, the diamond carat was formerly defined

by law to be 3.2 troy grains, which is about 207 milligrams.

Jewelers everywhere now use a metric carat defined in 1907 to

be exactly 200 milligrams or 0.2 gram.

38.
Volume for liquid

American Standard Metric Standard

1 gallon = 4 quarts 1 liter = 1000 milliliters

1 quart = 2 pints 1 milliliter = 1 c.c.

1 pint = 16 fluid ounces

(also 1 pint = 2 cups

1cup = 16 Tablespoons

1 TBS = 3 teaspoons )

Conversion:

1 gallon = 3.785 411 784 liters

3.8 liters

1 gallon = 231 cubic inches

American Standard Metric Standard

1 gallon = 4 quarts 1 liter = 1000 milliliters

1 quart = 2 pints 1 milliliter = 1 c.c.

1 pint = 16 fluid ounces

(also 1 pint = 2 cups

1cup = 16 Tablespoons

1 TBS = 3 teaspoons )

Conversion:

1 gallon = 3.785 411 784 liters

3.8 liters

1 gallon = 231 cubic inches

39.
Mr. Gallon is a visual aid for elementary school students to

remember the number of each unit that makes up one gallon

remember the number of each unit that makes up one gallon

40.
Do you know?

Fact: one c.c. of water weighs 1 gram at 4°C.

How much does one gallon of water weigh?

Answer: approx. 8.34 lb.

How much does one gallon of gasoline weigh?

Answer: varies between 5.8 to 6.5

depending on its type

Fact: one c.c. of water weighs 1 gram at 4°C.

How much does one gallon of water weigh?

Answer: approx. 8.34 lb.

How much does one gallon of gasoline weigh?

Answer: varies between 5.8 to 6.5

depending on its type

41.
Area

American Standard Historical Note

An acre is originally defined

1 square mile = 640 acres as the amount of land a pair

1 acre = 43560 square feet of oxen could plow in a day.

F.Y.I.

A football field (including end zones)

measures 57600 sq feet, hence it is equal to

1.322 acres, or approximately 1 and 1/3

acres.

American Standard Historical Note

An acre is originally defined

1 square mile = 640 acres as the amount of land a pair

1 acre = 43560 square feet of oxen could plow in a day.

F.Y.I.

A football field (including end zones)

measures 57600 sq feet, hence it is equal to

1.322 acres, or approximately 1 and 1/3

acres.

42.
Temperature

American standard Metric Standard

Fahrenheit Celsius

32 ºF = freezing 0 ºC = freezing

212 ºF = boiling 100 ºC = boiling

(for pure water) (for pure water)

9 5

F C 32 C ( F 32)

5 9

American standard Metric Standard

Fahrenheit Celsius

32 ºF = freezing 0 ºC = freezing

212 ºF = boiling 100 ºC = boiling

(for pure water) (for pure water)

9 5

F C 32 C ( F 32)

5 9

43.
Historical Note

The Fahrenheit scale was invented by German-born

scientist Gabriel Fahrenheit in 1714. He originally

defined the scale with 0 ºF representing the coldest

temperature he could create (in the hope of avoiding

negative numbers) with a mixture of ice and salt.

He also wanted 100 ºF to be about the human body

temperature, and wanted to have 180 equal parts

between the freezing- and boiling-points of pure

It turns out that the body temperature varies a lot

between people, and is not even constant for the same

person. The average is however, 98.6 ºF.

The Fahrenheit scale was invented by German-born

scientist Gabriel Fahrenheit in 1714. He originally

defined the scale with 0 ºF representing the coldest

temperature he could create (in the hope of avoiding

negative numbers) with a mixture of ice and salt.

He also wanted 100 ºF to be about the human body

temperature, and wanted to have 180 equal parts

between the freezing- and boiling-points of pure

It turns out that the body temperature varies a lot

between people, and is not even constant for the same

person. The average is however, 98.6 ºF.

44.
Section 13.2 Length and Area

45.
Addition of Distance

Addition Principle

The distance from point P to point Q, plus the distance from point Q to

point R is equal to the distance from point P to point R provided that Q

is on the segment PR.

P Q R

Triangle inequality

In general if Q is not on the segment PR, then

dist(PR) < dist(PQ) + dist(QR)

P

R

Q

Addition Principle

The distance from point P to point Q, plus the distance from point Q to

point R is equal to the distance from point P to point R provided that Q

is on the segment PR.

P Q R

Triangle inequality

In general if Q is not on the segment PR, then

dist(PR) < dist(PQ) + dist(QR)

P

R

Q

46.
Extended addition principle of distance

If the points B and C on the line segment AD, then

dist(AD) = dist(AB) + dist(CD) – dist(CB)

B

A

D

C

Two pipes, one 26 centimeters long, the other 18 centimeters long,

are joined to form a longer, straight pipe. To make the connection

secure, there is a 4 cm. overlap between the two pipes. How long is

the combined pipe?

If the points B and C on the line segment AD, then

dist(AD) = dist(AB) + dist(CD) – dist(CB)

B

A

D

C

Two pipes, one 26 centimeters long, the other 18 centimeters long,

are joined to form a longer, straight pipe. To make the connection

secure, there is a 4 cm. overlap between the two pipes. How long is

the combined pipe?

47.
Peri means “around” and meter means “measure”. Hence perimeter literally

means the measure all around.

The perimeter of a polygon is the sum of the lengths of its sides.

means the measure all around.

The perimeter of a polygon is the sum of the lengths of its sides.

48.
The perimeter of a circle is given the special name

In every circle, the ratio of the circumference C to the diameter

d is always the same number (the Greek letter “pi”).

This number is irrational, so it’s decimal representation is

non-terminating and non-repeating. The common

approximations are 3.1416 and 22/7.

In summary: C = d = 2r

In every circle, the ratio of the circumference C to the diameter

d is always the same number (the Greek letter “pi”).

This number is irrational, so it’s decimal representation is

non-terminating and non-repeating. The common

approximations are 3.1416 and 22/7.

In summary: C = d = 2r

49.
Area Formula for a Rectangle

The area of a rectangle is the (minimum) number of units

required to cover up the rectangle with no gaps and no

overlap.

unit

width

length

We see that the number of unit squares in a row is just the length, and the number

of units in a column is just the width, hence we have this simple formula:

Area = length × width

If we use rectangles or triangles as units, the formula will be more complicated.

The area of a rectangle is the (minimum) number of units

required to cover up the rectangle with no gaps and no

overlap.

unit

width

length

We see that the number of unit squares in a row is just the length, and the number

of units in a column is just the width, hence we have this simple formula:

Area = length × width

If we use rectangles or triangles as units, the formula will be more complicated.

50.
Relationship between Area and Perimeter of a

Rectangle

Can we have two rectangles with the same

area but different perimeters?

2

Can we have two rectangles with the same

perimeter but different areas? 6

Given a rectangle with perimeter 16 units,

what is its max area? Min area?

Given a rectangle of area 12 units, what is its 4

minimum perimeter? Max perimeter?

3

Rectangle

Can we have two rectangles with the same

area but different perimeters?

2

Can we have two rectangles with the same

perimeter but different areas? 6

Given a rectangle with perimeter 16 units,

what is its max area? Min area?

Given a rectangle of area 12 units, what is its 4

minimum perimeter? Max perimeter?

3

51.
Area formula for a Triangle

1

area base height

2

height

height height

base

base base

1

area base height

2

height

height height

base

base base

52.
Explanation for the Area formula for a

Right Triangle

1

area base height

2

First make a congruent copy.

Then rotate it 180°.

Next join the two right triangles

height

together, and we get …

A rectangle!

base

We divide the product by 2 b/c there are two right triangles in this rectangle.

Right Triangle

1

area base height

2

First make a congruent copy.

Then rotate it 180°.

Next join the two right triangles

height

together, and we get …

A rectangle!

base

We divide the product by 2 b/c there are two right triangles in this rectangle.

53.
The explanation for a scalene triangle is more tricky.

We begin with making a duplicate of the triangle (click)

We then cut the duplicate along the height (the vertical line starting from

the top of the triangle).

We begin with making a duplicate of the triangle (click)

We then cut the duplicate along the height (the vertical line starting from

the top of the triangle).

54.
Next we can maneuver the two smaller blue triangles to form a rectangle

with the original (red) triangle. (click)

height

base

Clearly the area of the rectangle is base×height, and the area of the red

triangle is consequently base×height/2.

with the original (red) triangle. (click)

height

base

Clearly the area of the rectangle is base×height, and the area of the red

triangle is consequently base×height/2.

55.
Area formula for an ObtuseTriangle

1

area base height

2

A parallelogram!

height

base

And we shall see very soon that the area formula for a

parallelogram is base × height.

1

area base height

2

A parallelogram!

height

base

And we shall see very soon that the area formula for a

parallelogram is base × height.

56.
Heron’s formula for Area of a Triangle

Given a triangle with sides a, b, and c, its

area is

a s( s a )( s b)( s c )

b

where s is the half-perimeter, i.e.

a bc

c s

2

Given a triangle with sides a, b, and c, its

area is

a s( s a )( s b)( s c )

b

where s is the half-perimeter, i.e.

a bc

c s

2

57.
Heron of Alexandria

58.
Area formula for a Parallelogram

height

base

Area = base × height

(Go to next page to see a proof.)

height

base

Area = base × height

(Go to next page to see a proof.)

59.
Area formula for a Parallelogram

height

base

Area = base × height

height

base

Area = base × height

60.
Area formula for a Trapezoid

a trapezoid is a quadrilateral with 2 (opposite) parallel sides.

top

height

bottom

top bottom

area ( ) height

2

a trapezoid is a quadrilateral with 2 (opposite) parallel sides.

top

height

bottom

top bottom

area ( ) height

2

61.
Let us find out why we have such a formula.

(1) we make a congruent copy of this trapezoid and slide it to the right.

top

height

bottom

(2) we rotate the copy 180 degrees.

(3) slide the up-side-down copy to the left so that it touches the original

(1) we make a congruent copy of this trapezoid and slide it to the right.

top

height

bottom

(2) we rotate the copy 180 degrees.

(3) slide the up-side-down copy to the left so that it touches the original

62.
Let us find out why we have such a formula.

(1) we make a congruent copy of this trapezoid and slide it to the right.

bottom

top

height

height

top

bottom

(2)Now

(4) we rotate the acopy

we have 180 degrees.

parallelogram whose base is (top + bottom), hence the

area of the parallelogram is

(3) slide the up-side-down copy to +the

(top left so ×that

bottom) it touches the original

height

trapezoid.

and the area of each trapezoid is

top bottom

area ( ) × height

2

(1) we make a congruent copy of this trapezoid and slide it to the right.

bottom

top

height

height

top

bottom

(2)Now

(4) we rotate the acopy

we have 180 degrees.

parallelogram whose base is (top + bottom), hence the

area of the parallelogram is

(3) slide the up-side-down copy to +the

(top left so ×that

bottom) it touches the original

height

trapezoid.

and the area of each trapezoid is

top bottom

area ( ) × height

2

63.
Area of a Circle

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

64.
Area of a Circle

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

65.
Area of a Circle

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

66.
Area of a Circle

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

the area of the following “parallelogram” is approx.

base × height = πr × r = πr2

height = r

half circumference = πr

If we cut the circle into a googol congruent sectors, the above figure

will become a rectangle, and its area will really be base × height .

The following animation gives an intuitive reasoning that

the area of a circle should be πr2.

the area of the following “parallelogram” is approx.

base × height = πr × r = πr2

height = r

half circumference = πr

If we cut the circle into a googol congruent sectors, the above figure

will become a rectangle, and its area will really be base × height .

67.
Addition properties of Area

If a region R is made up of two regions S and T that do not overlap (but

they can touch each other), then

area(R) = area(S) + area(T).

S

T

If a region R is made up of two regions S and T that do not overlap (but

they can touch each other), then

area(R) = area(S) + area(T).

S

T

68.
Find the area of the “key” of a standard basketball court.

12 ft 6 ft

19 ft

Answer: 284.5 ft2

12 ft 6 ft

19 ft

Answer: 284.5 ft2

69.
Pythagorean Theorem

Pythagoras was a Greek philosopher lived between (approx.) 582 BC and

507 BC.

This theorem was actually known to the Babylonians, Egyptians, and

Chinese more than 1000 years earlier, but Pythagoras was the first to

publish a proof.

Given any right triangle, if a and b are the

lengths of its two legs and c is the length of it

hypotenuse, then c

b

a 2 + b 2 = c2

a

Pythagoras was a Greek philosopher lived between (approx.) 582 BC and

507 BC.

This theorem was actually known to the Babylonians, Egyptians, and

Chinese more than 1000 years earlier, but Pythagoras was the first to

publish a proof.

Given any right triangle, if a and b are the

lengths of its two legs and c is the length of it

hypotenuse, then c

b

a 2 + b 2 = c2

a

70.
Pythagoras was also the first to publish a proof that the angle sum of a

triangle is always a straight angle, and there are only 5 regular

There are now more than 400 different proofs of the Pythagorean

In the history of America, there is only one president who made a

contribution to mathematics. His name James Garfield and he gave a

proof to the Pythagorean theorem in 1876. This was also his only

contribution to mathematics.

triangle is always a straight angle, and there are only 5 regular

There are now more than 400 different proofs of the Pythagorean

In the history of America, there is only one president who made a

contribution to mathematics. His name James Garfield and he gave a

proof to the Pythagorean theorem in 1876. This was also his only

contribution to mathematics.

71.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

In order to prove this, we first draw squares on each side of the triangle.

(click to see animation.)

c b

a

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

In order to prove this, we first draw squares on each side of the triangle.

(click to see animation.)

c b

a

72.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

In order to prove this, we first draw squares on each side of the triangle.

(click to see animation.)

c b

a

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

In order to prove this, we first draw squares on each side of the triangle.

(click to see animation.)

c b

a

73.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

Area = c2

c b Area = b2

Therefore we only have to

show that the area of the a

green square and area of We then calculate the area of

the blue square add up to Area = a2 each square. (click)

that of the yellow square.

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

Area = c2

c b Area = b2

Therefore we only have to

show that the area of the a

green square and area of We then calculate the area of

the blue square add up to Area = a2 each square. (click)

that of the yellow square.

74.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

Area = c2

Area = b2

We first cut the green

square into 4 congruent

pieces. (click) The second cut is perpendicular

The first cut is parallel to to the first but still through the

Area = a2 center.

the hypotenuse of the

triangle and through the When you are ready, click to see

center of the square. the assembling of the pieces

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

Area = c2

Area = b2

We first cut the green

square into 4 congruent

pieces. (click) The second cut is perpendicular

The first cut is parallel to to the first but still through the

Area = a2 center.

the hypotenuse of the

triangle and through the When you are ready, click to see

center of the square. the assembling of the pieces

75.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

76.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

77.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

78.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

79.
Pythagorean Theorem: Given a right triangle, if a and b are the length

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

Now you see that a2 + b2 is really equal to c2 !

of its two legs, and c is the length of the hypotenuse, then a2 + b2 = c2 .

Now you see that a2 + b2 is really equal to c2 !

80.

81.

82.
Figures on a Geoboard

Geoboard is a square arrangement of pegs where rubber bands are held in

place by the pegs.

It is a great tool for exploring a variety of mathematical topics introduced

in the elementary and middle schools. Learners stretch rubber bands

around the pegs to form line segments and polygons and make

discoveries about perimeter, area, angles, congruence, fractions, etc.

Geoboard is a square arrangement of pegs where rubber bands are held in

place by the pegs.

It is a great tool for exploring a variety of mathematical topics introduced

in the elementary and middle schools. Learners stretch rubber bands

around the pegs to form line segments and polygons and make

discoveries about perimeter, area, angles, congruence, fractions, etc.

83.
Dissection Method

This method is also called the “addition method”. It allows us to find

the area of many polygonal figures by cutting it into several smaller but

simpler pieces. Sometimes it even works for non-polygonal figures,

such as semicircles.

Note: each vertex of each piece must be on a peg.

This method is also called the “addition method”. It allows us to find

the area of many polygonal figures by cutting it into several smaller but

simpler pieces. Sometimes it even works for non-polygonal figures,

such as semicircles.

Note: each vertex of each piece must be on a peg.

84.
Subtraction Method

In some cases, there is simply no way to cut the figure into simple

pieces such that every vertex is on a peg. In those cases, we can

construct a rectangle outside the figure and then subtract areas of parts

not belonging to the figure.

Note: each vertex of each piece must be on a peg.

In some cases, there is simply no way to cut the figure into simple

pieces such that every vertex is on a peg. In those cases, we can

construct a rectangle outside the figure and then subtract areas of parts

not belonging to the figure.

Note: each vertex of each piece must be on a peg.

85.
Section 12.6

Describing 3D Shapes

Describing 3D Shapes

86.
Polyhedra

A polyhedron is the union of polygonal regions, any two of which have at

most one side in common, such that a connected finite region in space is

enclosed without holes.

These are polyhedra.

These are not polyhedra.

A polyhedron is the union of polygonal regions, any two of which have at

most one side in common, such that a connected finite region in space is

enclosed without holes.

These are polyhedra.

These are not polyhedra.

87.
A polyhedron is convex if every line segment joining two of its points

lies completely inside the polyhedron, or is on one of the polygonal faces.

Vertex

Face

Edge

lies completely inside the polyhedron, or is on one of the polygonal faces.

Vertex

Face

Edge

88.
Polyhedra can be classifed into several general types.

Prisms

are polyhedra with two opposite faces that are identical polygons. These

faces are called bases. The vertices of the bases are joined to form lateral

faces that must be parallelograms.

If the lateral faces are rectangles, the prism is called a right prism.

Otherwise it is called an oblique prism.

Prisms

are polyhedra with two opposite faces that are identical polygons. These

faces are called bases. The vertices of the bases are joined to form lateral

faces that must be parallelograms.

If the lateral faces are rectangles, the prism is called a right prism.

Otherwise it is called an oblique prism.

89.
Fingal's Cave, Island of Staffa, Scotland

Columnar Basalts are hexagonal prism rock

formations resulting from quick cooling of lava flow.

Columnar Basalts are hexagonal prism rock

formations resulting from quick cooling of lava flow.

90.
Pyramids

are polyhedra formed by using a polygon for the base and a point not on

the base as the apex, that is connected with line segments to each vertex

of the base.

If the base is a regular polygon, and all those lateral faces are isosceles

triangles, then it is called a right regular pyramid.

If the base is a regular polygon but the lateral sides are not iscosceles

triangles, it is called an oblique regular pyramid.

are polyhedra formed by using a polygon for the base and a point not on

the base as the apex, that is connected with line segments to each vertex

of the base.

If the base is a regular polygon, and all those lateral faces are isosceles

triangles, then it is called a right regular pyramid.

If the base is a regular polygon but the lateral sides are not iscosceles

triangles, it is called an oblique regular pyramid.

91.
Regular polyhedron

is a polyhedron in which all faces are identical regular polygons and

all dihedral angles have the same measure.

A dihedral angle is the angle formed by two adjacent faces of the

There are only five regular polyhedra, they are called the Platonic

is a polyhedron in which all faces are identical regular polygons and

all dihedral angles have the same measure.

A dihedral angle is the angle formed by two adjacent faces of the

There are only five regular polyhedra, they are called the Platonic

92.
Euler’s Formula for Polyhedra

F+V–E=2

F = the number of faces

V = the number of vertices

E = the number of edges

F+V–E=2

F = the number of faces

V = the number of vertices

E = the number of edges

93.
A cone is the union of a simple closed curve and all line segments

joining points of the curve to a point, called the apex, which is not in

the plane of the curve.

The plane curve together with its interior is called the base (of the

joining points of the curve to a point, called the apex, which is not in

the plane of the curve.

The plane curve together with its interior is called the base (of the

94.
A sphere is defined as the set of all points in three dimensional space

that are the same distance from a fixed point, called the center.

that are the same distance from a fixed point, called the center.

95.
Spheres

A sphere is defined as the set of all points in three dimensional space

that are the same distance from a fixed point, called the center. The

segment connecting any point of the sphere to the center is called a

radius of the sphere.

Great Circle of a Sphere

A great circle of a sphere is the intersection of the sphere and a plane

that passes through the center point of the sphere.

It is also one of the largest circle you can draw on that sphere.

If the radius of the sphere is R, then the

radius of any great circle is also R.

A sphere is defined as the set of all points in three dimensional space

that are the same distance from a fixed point, called the center. The

segment connecting any point of the sphere to the center is called a

radius of the sphere.

Great Circle of a Sphere

A great circle of a sphere is the intersection of the sphere and a plane

that passes through the center point of the sphere.

It is also one of the largest circle you can draw on that sphere.

If the radius of the sphere is R, then the

radius of any great circle is also R.

96.
Sphere Packing Problem

Since spheres are round, there is no way to put spheres in a box

without leaving any space.

So the problem is: what is the most efficient way to put identical

spheres is a large box?

The obvious way is the way that

greengrocers all over the world

would stack oranges, which is called

the face-centered cubic lattice

However, it took mathematicians

more than 300 years to prove it.

Since spheres are round, there is no way to put spheres in a box

without leaving any space.

So the problem is: what is the most efficient way to put identical

spheres is a large box?

The obvious way is the way that

greengrocers all over the world

would stack oranges, which is called

the face-centered cubic lattice

However, it took mathematicians

more than 300 years to prove it.

97.
The Japanese came up with a clever way to pack more watermelons

into a box. They made the watermelon a cube!

Cubic Watermelon! USD$300 each when it first appeared in market,

now it is about $83.

into a box. They made the watermelon a cube!

Cubic Watermelon! USD$300 each when it first appeared in market,

now it is about $83.

98.
This is the glass mold for

the new cubic/square

the new cubic/square

99.
Section 13.3 Surface Area

100.
Section 13.3 Surface Area

The surface area of a three-dimensional figure is the total area of its

exterior surfaces.

Surface area of a rectangular box

If ℓ = length, w = width, and h = height, then

surface area = 2 (ℓ ×w) + 2 (ℓ ×h) + 2(w×h)

h

ℓ

w

The surface area of a three-dimensional figure is the total area of its

exterior surfaces.

Surface area of a rectangular box

If ℓ = length, w = width, and h = height, then

surface area = 2 (ℓ ×w) + 2 (ℓ ×h) + 2(w×h)

h

ℓ

w

101.
Section 13.3 Surface Area

Nets for 3D solids

A net is a two dimensional plan or shape that can be folded to make a

3 dimensional solid. For some solids, such as that cube, can have

many different nets.

Cutting out these nets, folding and

gluing them to create a solid

object, will help children become

familiar with the features of these

solids. Tabs are sometimes

included to help gluing the faces

together, they are not part of the

net.

Nets for 3D solids

A net is a two dimensional plan or shape that can be folded to make a

3 dimensional solid. For some solids, such as that cube, can have

many different nets.

Cutting out these nets, folding and

gluing them to create a solid

object, will help children become

familiar with the features of these

solids. Tabs are sometimes

included to help gluing the faces

together, they are not part of the

net.

102.
Surface area of a prism

If h is the height, A is the area of each base, and P is the perimeter

of each base, then the surface area of a right prism is

S = 2A + P× h

h

a a

a

3 3 2

area of regular hexagon a

2

If h is the height, A is the area of each base, and P is the perimeter

of each base, then the surface area of a right prism is

S = 2A + P× h

h

a a

a

3 3 2

area of regular hexagon a

2

103.
Surface area of a right circular cylinder

If r is the radius and h is the height, then the surface area of the

right circular cylinder is

S = 2 π r 2 + (2 π r) h

r

h

If r is the radius and h is the height, then the surface area of the

right circular cylinder is

S = 2 π r 2 + (2 π r) h

r

h

104.
Surface area of a Right Regular Pyramid

If A is the base area, P is the perimeter of the base, and ℓ is the slant

height, then the surface area is

S=A+½Pℓ

The red line segment is

the slant height.

a

a2

Area of a regular pentagon is 25 10 5

4

If A is the base area, P is the perimeter of the base, and ℓ is the slant

height, then the surface area is

S=A+½Pℓ

The red line segment is

the slant height.

a

a2

Area of a regular pentagon is 25 10 5

4

105.
Surface area of a Right Circular Cone

If r is the radius of the base, h is the height, and ℓ is the slant height,

then the surface area is

S = π r 2 + ½ (2π r) h 2 r 2

= πr2+ πrℓ

ℓ

r

ℓ

A net for the cone.

If r is the radius of the base, h is the height, and ℓ is the slant height,

then the surface area is

S = π r 2 + ½ (2π r) h 2 r 2

= πr2+ πrℓ

ℓ

r

ℓ

A net for the cone.

106.
Spheres

Great Circle of a Sphere

A great circle of a sphere is the intersection of the sphere and a plane

that passes through the center point of the sphere.

It is also one of the largest circle you can draw on that sphere.

If the radius of the sphere is R, then the

radius of any great circle is also R, and the

area of each great circle will be R2

The surface area of the sphere will be 4

times the area of any great circle, i.e. 4R2

.

Great Circle of a Sphere

A great circle of a sphere is the intersection of the sphere and a plane

that passes through the center point of the sphere.

It is also one of the largest circle you can draw on that sphere.

If the radius of the sphere is R, then the

radius of any great circle is also R, and the

area of each great circle will be R2

The surface area of the sphere will be 4

times the area of any great circle, i.e. 4R2

.

107.
Surface area of a sphere

Area of Sphere = 4π r2

Since the area of a great circle in a sphere is πr2 , this formula says that

the surface area of a sphere is 4 times the area of any of its great

A sphere is frequently approximated

by a truncated icosahedron, such as a

soccer ball.

Here is a net of an icosahedron,

and in the lab, we will use its

surface area to approximate that

of a sphere.

Area of Sphere = 4π r2

Since the area of a great circle in a sphere is πr2 , this formula says that

the surface area of a sphere is 4 times the area of any of its great

A sphere is frequently approximated

by a truncated icosahedron, such as a

soccer ball.

Here is a net of an icosahedron,

and in the lab, we will use its

surface area to approximate that

of a sphere.

108.
Surface area of a sphere

Archimedes observed (without

explanation) that the sphere has the

same area as the cylinder with the same

diameter and same height.

(In this picture, part of the cylinder is

removed to show the sphere)

The diameter is 2r, hence the height is

also 2r.

The circumference of the cylinder is

2π r,

Hence the area of the cylinder is

2r × 2π r = 4π r2

However, this argument is not true

Area of Sphere = 4π r2

for shapes other than a sphere.

Archimedes observed (without

explanation) that the sphere has the

same area as the cylinder with the same

diameter and same height.

(In this picture, part of the cylinder is

removed to show the sphere)

The diameter is 2r, hence the height is

also 2r.

The circumference of the cylinder is

2π r,

Hence the area of the cylinder is

2r × 2π r = 4π r2

However, this argument is not true

Area of Sphere = 4π r2

for shapes other than a sphere.

109.
Section 13.4 Volume

- the amount of space that a substance or

object occupies, or that is enclosed within

a container.

- the amount of space that a substance or

object occupies, or that is enclosed within

a container.

110.
Volume of a Rectangular Solid

V=ℓ×w×h

h

ℓ

w

V=ℓ×w×h

h

ℓ

w

111.
Volume of a Prism

V=A×h where A is the area of the top (or base).

a a

a

3 3 2

area of regular hexagon a

2

where a is the length of each side

V=A×h where A is the area of the top (or base).

a a

a

3 3 2

area of regular hexagon a

2

where a is the length of each side

112.
Volume of a Cylinder

V = π r2 × h

V = π r2 × h

113.
Cavalieri’s Principle

Suppose that two 3D solids have the same height and are such that

every plane parallel to the base cut cross-sections of the solids with

equal areas. Then the volume of the solids are equal.

Suppose that two 3D solids have the same height and are such that

every plane parallel to the base cut cross-sections of the solids with

equal areas. Then the volume of the solids are equal.

114.
Generalized Cavalieri’s Principle

Suppose that two 3D solids A and B are contained between two

parallel planes. If at any level, the area of the cross-section of solid A

is always k times that of solid B, then

volume of solid A = k × (volume of solid B)

Suppose that two 3D solids A and B are contained between two

parallel planes. If at any level, the area of the cross-section of solid A

is always k times that of solid B, then

volume of solid A = k × (volume of solid B)

115.
Volume of a Square Pyramid

The following pyramids have congruent square bases and both have

heights equal to the width of the base.

And since they have equal cross-sectional areas, they have the same

Right square pyramid Three congruent copies of

this pyramid form a cube.

1

Hence the volume of this pyramid is base area × height.

3

The following pyramids have congruent square bases and both have

heights equal to the width of the base.

And since they have equal cross-sectional areas, they have the same

Right square pyramid Three congruent copies of

this pyramid form a cube.

1

Hence the volume of this pyramid is base area × height.

3

116.
Volume of a Pyramid

1

V A h

3

a a

a

“A” is the area of the base.

a2

Area of a regular pentagon is 25 10 5

4

1

V A h

3

a a

a

“A” is the area of the base.

a2

Area of a regular pentagon is 25 10 5

4

117.
Volume of a Cone

If the base is a circle with radius r and the height is h, then

1

V π r 2 h

3

If the base is a circle with radius r and the height is h, then

1

V π r 2 h

3

118.
Volume of a Sphere

Archimedes “proved” that the sphere has the same volume as

the solid on the right, which has the same diameter and height

as the sphere.

4

V π r 3

3

part of this solid is cut away to

show the interior.

Archimedes “proved” that the sphere has the same volume as

the solid on the right, which has the same diameter and height

as the sphere.

4

V π r 3

3

part of this solid is cut away to

show the interior.