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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.
2.
Section 13.1
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
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.
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.
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.
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.
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
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.
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
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.
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.
13.
Measuring the length of a curve with paper clips
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.
15.
How to tile a rectangular floor?
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
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)
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?
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.
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?
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.
22.
Common Standard Units
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.
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.
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
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
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.
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)
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.
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
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
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.
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
34.
(Go to the next page)
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.
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.
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
39.
Mr. Gallon is a visual aid for elementary school students to
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
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.
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
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.
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
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?
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.
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
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.
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
51.
Area formula for a Triangle
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.
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).
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.
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.
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
58.
Area formula for a Parallelogram
height
base
Area = base × height
(Go to next page to see a proof.)
59.
Area formula for a Parallelogram
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
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
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
63.
Area of a Circle
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.
65.
Area of a Circle
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 .
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
68.
Find the area of the “key” of a standard basketball court.
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
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.
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
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
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.
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
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 .
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 .
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 .
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 .
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 !
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.
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.
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.
85.
Section 12.6
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.
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
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.
89.
Fingal's Cave, Island of Staffa, Scotland
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.
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
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
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
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.
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.
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.
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.
98.
This is the glass mold for
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
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.
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
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
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
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.
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
.
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.
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.
109.
Section 13.4 Volume
- 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
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
112.
Volume of a Cylinder
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.
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)
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
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
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
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.
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