Download Chapter I Getting Your Bearings, The Sizes of Things 1 Chapter I

Chapter I Getting Your Bearings, Math Skills and The Sizes of Things
Finding sizes: As part of the first assignment, you will be finding sizes of things. You might need to
find mass or radius or lifetime. The assignment is geared to your textbook, although you may use the
internet to find the information. Be very careful to get what is requested. Download and save the
Internet reference if you want to be able to show that the answer is correct.
Part of what we are learning is how to convert units and how to use a logarithmic plot. So don’t spend a
lot of time searching for the exact units (e.g. finding meters vs Earth radii).
Converting Units: We compare the sizes of objects in order to get a concept of the Universe. The term
size is somewhat vague. It might mean mass, radius, area, circumference, duration in time etc.
Generally these features are not interchangeable. They have different meanings and refer to different
features.
Even when we have comparable measurements e.g. lengths, the values all need to be in the same
units so that we can understand what is larger. If the information isn’t all in the same units, it is
necessary to convert it so that they can be compared. What if we don’t have the information in the
same units? It is necessary to convert them to the same units.
For the first homework, we will find a dimension: that is, a length, width, diameter etc. Other forms of
size, such as mass, area, or volume are useful, but not directly comparable. To decide whether you
have found the proper dimension, the values must be in units of length e.g. meters, miles, kilometers
etc.
There may not be one exact answer for each item. For example, you might be seeking the size of an
automobile. Possible answers would be the length of a Mini Cooper, about 97 inches long
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0
(2.46x10 meters in scientific notation) and 55.4 inches (1.47 x10 meters) high. A Hummer H3 might be
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6.7x10 meters long and about 1.85 meters high. We can convert the lengths or heights of the vehicles
to the same units to see which is larger. On the other hand, the mass of the auto, or the surface area
are different from one another and different from the length or height. They cannot be compared
directly. They need to be in the same units.
Units In science (and almost everywhere but the USA) the metric system is used. So lengths should
be in centimeters, meters, or kilometers. Time is usually in seconds. Mass would be in grams or
kilograms. In astronomy there are some other, unique, units you will be seeing. These include
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Astronomical Unit – distance between the Earth and Sun=1.496x10 km=1.496x10 m
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15
Light Year(ly), distance light travels in a year=9.46x10 km=9.46x10 m
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Parsec(pc) = 3.26 light years= 206265 Astronomical Units=3.0856x10 km=3.0856x10 m
(a parsec is the distance of a body whose parallax is 1 second of arc)
Your book has quite a few relations between units (appendix and inside front cover). For example
100 centimeters (cm)=1 meters (m)
1000 meters (m) = 1 kilometers (km)
1 mile (mi)= 1.609 km
1 meter = 39.37 inches (beware, meter is abbreviated m and mile is abbreviated
mi)
It is handy to write down all the conversion factors in one place, like a page of your notebook, so you
don’t need to search..
Converting units does not change their meaning. But, as you know from algebra class, the only things
that can be done to a number without changing its value are to add 0 or to multiply by 1. To convert
multiply by 1.
This may sound useless, how can multiplying by 1 do anything at all? To convert units, you might use
any of the following. They are all equal to 1, since they are the same on the top and bottom.
Chapter I Getting Your Bearings, The Sizes of Things
1
3 9.4605x10 15 m
3 1 light year
1.4960x1011 m
1AU
3.0857x10 16 m
1km
1 in
4
1 pc
3.937x10 in .0254 m
All of these values are equally true, but each is most useful when the units of the denominator (bottom
of the fraction) are the same as the units of the value you want to convert, so that the units cancel.
Read through the examples to see how it works.
€
Example: Saturn’s orbit has a semimajor axis of 9.3539 AU. How large is the semimajor axis in
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meters? To get from AU to meters, look up 1 AU = 1.496x10 m
Since these values are equal, they can be placed, one over the other, to make a form of 1. How should
it be done? The units of the previous value (the AU) should be on the bottom, to cancel.
9.539 AU
Cancel the AU to get
⎛1.496 x 1011 m ⎞
= 9.539AU x ⎜
⎟
1 AU
⎝
⎠
= 9.539 x1.496 x 1011 m
= 1.427x1012 m
Example: Sirius is 2.7 parsecs away from the Sun. How far away is this in meters?
€
The parsecs cancel.
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Example: The distance to Vega is 2.39x10 m. How many light years is that?
⎛ 1 light year ⎞
⎟ , with the meters on the bottom to cancel the units. So
⎝ 9.46x1015 m ⎠
Multiply the distance by ⎜
⎛ 1 light year ⎞
2.39 × 1017 m × ⎜
⎟
⎝ 9.46 × 1015 m ⎠
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€ year ⎞
⎛ 2.39 × 1017 light
Since 9.46x10 was on the bottom of the fraction, we divide BY it to get
= ⎜
⎟
15
9.46 × 10
⎝
⎠
= 25.26 light years
€
When you start, there is no equation. You just write the number and equate it to itself times ONE. You
can tell that the conversion is correct if the units (just the names like parsec or km) cancel top and
bottom. Find these “same” values from a textbook or the work book. You do not need to memorize the
number values.
Often, you will not find a single equation relating the original units to the final value. In that case, find
relations between the current units and some other, then between that unit and another, etc. etc. until
you have steps relating one unit to the next, without skipping any steps. In this case you will be
multiplying by 1 several times. It is generally better to write out ALL the terms, cancel the units (to be
certain that the correct values are being used) and only then to multiply all the values together.
Example: A football field is 300 feet. How many kilometers is that? Find the conversion factors to go
from the original units, step by step, to the units you want.
⎛ 12inches ⎞ ⎛ 0.0254m ⎞ ⎛ 1km ⎞
300feet × ⎜
⎟ × ⎜
⎟ × ⎜
⎟
⎝ 1foot ⎠ ⎝ 1inch ⎠ ⎝1000m ⎠
⎛ 0.0254m ⎞ ⎛ 1km ⎞
= 3600 inches × ⎜
⎟ × ⎜
⎟
⎝ 1inch ⎠ ⎝ 1000m ⎠
⎛ 1km ⎞
= 91.44 × ⎜
⎟
⎝1000m ⎠
= 0.09144km
Check yourself: Convert each of the following a) 1.7x10
Convert 13 Mpc to m d) 14kpc to m e) 3.4 ft to m
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inches to m
b) 14 yards to miles c)
€
Chapter I Getting Your Bearings, The Sizes of Things
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b) 14yd x⎛⎜ 3ft ⎞⎟ x⎛⎜ 1mi ⎞⎟ = 7.95x10 −3 mi c)
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a) 1.7x10 inches=4.318x10 m
⎝ 1yd ⎠ ⎝ 5280ft ⎠
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d) 14 kpc = 4.326x10 m
0
e) 3.4 ft = 1.04x10 m
€
Calculator Note: The order of multiplication and division doesn’t matter. On the other hand, many
calculators will need more parentheses than you might think.
Example: If you type 6/5x7 into a calculator, you get 8.4 not 0.171. The calculator will divide 6 by 5
and then multiply the answer by 7. To divide 6 by 5x7, you need to type 6/(5x7) or 6/5=/7=.
Scientific Notation; This is a style of numbers. Scientific notation does not change the information, but
does make it easier to compare values and eliminates the need to write out many zeroes.
In Scientific Notation the value is written as
a number between 1 and 9.9999 times an integer power of 10.
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So the following are written in scientific notation: 5x10 , 4.23x10 , 1.09 x 10 while the following are not:
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50, 43.9x10 , 0.00109 x 10 although these numbers represent the same values. When operating with
scientific notation, the easiest thing is to load the values (in whatever format) into your calculator, find
the answer, and then write the answer in scientific notation.
To put a number into scientific notation, move the decimal point until it is to the right of the first non-zero
digit and count how many places you have moved it. Then multiply by 10 to the power equal to the
number of places that you moved the decimal. If you moved the decimal to the left, making the number
smaller, then multiply by 10 to a positive power. If you moved it to the right, making the number larger,
then multiply by 10 to a negative power. If there is no decimal showing, it is past the last digit to the
right. For example
5234.=5.234x10
0.0034=3.4x10
3
-3
Calculator Note: Calculators express the power of 10 using a key that says EE or EXP or ee . The ee
key is usually a second function (above the actual keys). These keys tell the calculator that the next
number will be the power of 10 that is part of the number.
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e.g. 5.234x10 is 5.234 EE 3 or 5.234 EXP 3
-3
3.4x10 is 3.4 EE -3 When you want to enter the -3, it is necessary to use the +/- key.
Things that don’t give the right answer include:
a) typing x10 as part of the number
x
b) using the e key or e key for a power of 10
c) using the - key to get a negative number.
Plotting: We want to plot the sizes of astronomical objects to compare them. The wide range of
sizes doesn’t usually fit on a single plot, so we use a logarithmic plot, like the following.
A logarithmic plot gives equal space to equal powers of 10, rather than giving equal space to equal
size intervals. The major divisions start at 1 times a power of 10. Numbers that are not exactly powers
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of 10 (e.g. 4 x 10 , 3.8 x10 etc) are plotted between the major divisions. The tiny numbers 2, 3,5,8
show which mark to use for 2 times the power of 10 that is immediately below, 3 times, etc. The tic
marks show all of the multipliers (2x, 3x, 4x etc). They are not all labeled because there is too little
space.
To use the graph, each number must be in scientific notation first (a value between 1 and 9.999 times
an integer power of 10). Since the number is LARGER than the power of 10, its value will be plotted
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ABOVE the power of 10, at the position indicated by the leading number. For example 8x10 km
Chapter I Getting Your Bearings, The Sizes of Things
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Sizes in Meters
1.0x10 0
1.0x10-1
1.0x10-2
1.0x10-3
1.0x10-4
1.0x10-5
1.0x10-6
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
1.0x10-7
8
5
3
2
1.0x10-8
8
5
3
2
1.0x10-9
8
5
3
2
1.0x10-10
1.0x10-11
1.0x10-12
1.0x10-13
1.0x10-14
1.0x10-15
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
1.0x1015
1.0x1014
1.0x1013
1.0x1012
1.0x1011
1.0x1010
1.0x109
1.0x108
1.0x107
1.0x106
1.0x105
1.0x104
1.0x103
1.0x102
1.0x101
1.0x10 0
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
1.0x10 30
1.0x10 29
1.0x1028
28
282
1.0x1027
1.0x1026
1.0x1025
1.0x1024
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
1.0x1023
8
5
3
2
8
5
3
2
1.0x1022
8
5
3
2
8
5
3
2
1.0x1021
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
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Chapter I Getting Your Bearings, The Sizes of Things
1.0x1020
1.0x1019
1.0x1018
1.0x1017
1.0x1016
1.0x1015
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
8
5
3
2
4
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appears above the 1x10 line at the value 8. The value 1.2x10
19
but below the 2x10 line.
19
appears slightly above the 1x10
19
line,
When you plot the values you have found on the logarithmic graph provided and label them with the
name of the object. The vertical coordinate on the plot is the size of the object in kilometers. The
horizontal coordinate on this graph has no meaning (although it does on other graphs).
Theory Note: How is spacing of the numbers decided? Every positive number can be represented by
10 raised to some power. This power is called the logarithm of the number. The power is not usually an
integer. The spacing is determined by this power. So, for example,
log (4)
.602
4=10
=10
, so the 4 is about 0.6 of the way from the bottom of the interval to the top. The
spacing of the small marks takes the logarithms into account.
Math Notes
Order of Operations
Do whatever is within parentheses first. Then:
Raise to powers (i.e. evaluate the effect of the exponent)
Multiply and divide (order doesn’t matter)
Add and Subtract (order doesn’t matter between these)
So, for example
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2
6 + 9 +4x5/17 = ?
would be evaluated as follows.
Do powers first, then multiplication and division, then addition and subtraction.
(63) + (92)+{4x(5/17)} or (63)x(92)+{(4x5)/17}
=216 + 81 +20/17= 297+20/17 =297+1.176=298.176
When using a calculator, be prepared to use parentheses in more places that you might expect. For
example, the expression, 6x 7 should come out as 3.5. The order of multiplication and division doesn’t
4x 3
matter, but the expression includes dividing by 3. A calculator may give (6x7/4)x3 = 31.5, depending on
how the information is entered. The calculator needs to be told explicitly to divide by 3.
Exponential Notation
The symbols 103, 1E3, and 10**3 all mean 10 to the third power i.e. 10 x 10 x 10. The first way of
writing is used if the writer can do superscripts, the other ways are used in some computer notations
where superscripts are not available
More generally 10n means 10 multiplied together n times,
i.e. 10x10x10...x10 where there are n repetitions of 10
(or equivalently 1 with n zeroes following it.).
To multiply together two numbers of the form 10n x 10m, the exponents are added, e.g.
10n x 10m=10m+n
Similarly, 145 means 14 x 14 x 14 x 14 x 14 = 537824 not 14x105.
A negative exponent, means 1 divided by the number raised to the power given. e.g.
10-1 means 1/10
2
2-2 means 1/2 =1/4
10-2 means (1/10) multiplied together twice, i.e.
2
10-2 =1/10 = (1/10) x (1/10) = 0.01
-4
Similarly,
17-3= 1/(17 x 17 x 17) = 2.035x10 = 0.0002035.
Chapter I Getting Your Bearings, The Sizes of Things
5
The rule of adding the exponents when multiplying numbers of the form 10-n x 10m holds so
10-n x 10m =10-n+m =10m-n
or subtract the exponent on the bottom when dividing. For example
m
n
m-n
10 /10 =10
In math, two negatives cancel (or two wrongs make a right, if you like), so
m
m – (-n)
m +n
-n
10 /10 =10
=10
Examples
109/ 1012 = 10 9-12 = 10 –3
109/10-12 = 109-(-12) =1021
Addition/Subtraction The calculator will take any combination, but there is no simple way to add
exponential numbers, by hand. You need to either write them out explicitly or factor them e.g.
102+104= 100+10000=10100
=102 x (1+102)=102 x (1+100)= 102 x (101)=10100
Computing with Scientific Notation and Similar Forms
Because the order of multiplication and division doesn’t matter, scientific notation can be used in
computation as follows.
6
9
6+9
15
16
3.1x10 x 7.5x 10 = 3.1 x 7.5 x 10
= 23.25 x 10 =2.325 x 10 in scientific notation. It is
necessary to increase the exponent to compensate for decreasing the value 23.25 to 2.325 and
increasing the exponent.
It is NOT necessary to separate the powers of 10 from the rest of the computation. It is helpful if you
are doing the computation by hand, because it is easy to compute the powers of 10 and estimate the
rest. In the computation about, the 3x7 part could be estimated as 21. That would not be final answer,
but it would give a rough idea of the result.
Solving Equations
Often we have a formula where we know values for some, but not all, of the variables. For example
Distance =Rate xTime
Distance= 340 kilometers, Time= 3 hours
The goal is to rearrange the values so that an unknown value appears on one side, alone, and only
known values appear on the other. The way to do this without destroying the information embodied in
the equation is to use
EQUAL TREATMENT
for both sides of the equation
You can do the following without making an equation untrue:
Substitute a numerical value for a variable, if you know the value
Multiply or Divide by something, both sides (using zero here is legal, but it destroys the
process)
Add or Subtract something, both sides
Raise something to a power (like square it, take the square root)
Clear parentheses, for example a(b+c) = ab + bc
and you should do one or another of the above (or several) if it helps to get all the known things on one
side, and the unknown things on the other.
Looking back at the problem,
Distance =Rate xTime
Distance = 340 kilometers, Time= 3 hours
Chapter I Getting Your Bearings, The Sizes of Things
6
We know the distance and the time and want to find the rate. So divide both sides by Time, to get rate
by itself
(1/Rate) x Distance =Rate x Time x (1/Rate)
The point was to be able to cancel the Rate, so we do to get
(1/Rate) x Distance =Rate x Time x (1/Rate)
Distance/ Time = Rate
Now it is easy to substitute, (Distance = 340 kilometers, Time= 3 hours)
340 km/3 hr = Rate
113.33 km/ hr = Rate
Algebra Advice
Solutions in algebra are not very intuitive. Don’t expect to know the answer automatically, no matter
how brilliant you are. It is rather like fixing a car or cooking. No matter how smart you are, it still takes all
the steps and a bunch of time. It doesn’t mean that you are dumb or bad at math. It just works slowly
and systematically. It is best to write out EVERY step completely. Be sure that each line is a repetition
of a true equation. That way, you can understand what you have done if you come back.
Not every legal thing you may do will help isolate the unknown. If it doesn’t help, it may be necessary
to start again. Even if it seems to have helped, it may be worthwhile to put the problem away.
A good way to check your work is to put the answer aside, separate from the problem. Come back to it
when you have not thought about it for a while. Redo the problem. NOW compare your results. If they
are different, at least one must be incorrect. If you are working with a friend, compare methods, each
reading the other person’s. Or read the work out loud to someone, even if they have not done the
problem. It will make you go over the information slowly and often help you to notice an error. It is hard
to read your own work and find the error. We all want to be right and that gets in our way as we look at
our own results.
Uncertainty and Number of Significant Figures
Most numbers we use are the results of measurements. That means that they are not perfectly
accurate. Unfortunately not all textbooks tell you how accurate the values are and it is not true that the
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last digit is the only one that is uncertain, Consider a value like 3.88x10 . The range of values really
-9
-9
might really be from 4.03 x10 to 3.5x10 with probability 95% (typically the exponent would not
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change). Another way that this can be written is 3.88 x10 ( +0.15, -0.38, 2σ). There is no way for you
to know the uncertainty is unless the author specifies it. In this case, we would say that the value has 1
significant figure, since the range of values from largest to smallest leaves only one digit unchanged.
If we don’t have detailed information about the measurement uncertainty, the number of significant
figures is used to estimate the error. When computing with measurement values, you should get a
result only as good as the LEAST accurate of the numbers included. If the only operations are
multiplication and division, and the values are all expressed in scientific notation, the number of figures
in a number is the number of digits in the number before the power of 10. The power of 10 does NOT
affect the number of significant figures. If the number is NOT in scientific notation, leading zeroes do not
count as significant figures, but trailing zeroes do. Examples:
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0.09 has 1 significant figure (9. x 10 )
0.010 has 2 significant figures since the 0 following the 1 indicates that you know the next digit. You
-2
could write it as 1.0 x 10 . The leading zero(s) indicate the power of 10, but is (are) not considered to
be significant figures
9
9.87 x 10 has 3 significant figures
9
Thus, the result 0.09 x 0.010/9.87x10 = 9x10
the least accurate value in the computation.
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should have ONLY 1 significant figure, the same as
Chapter I Getting Your Bearings, The Sizes of Things
7
If addition or subtraction is involved in a computation, the numbers need to be written out, and digits
should be kept so long as they are the result of known values.
3
Example 3.24 x 10 +204+62.6=
Adding 3240 ( actually we don’t know the 0, but it is needed to keep the columns straight)
204
62.6
3506.6 This value does not include the correct number of significant figures. If we write it in
3
3
scientific notation the result becomes 3.5066x10 , with all the digits, and 3.51 x10 with the correct
number of digits. The two 6’s resulted from addition of some uncertain values. The final answer was
rounded up to get the best value
Conversion Factors
1 meter = 39.37 in (meter is abbreviated m)
1 foot = 12 in
1 in = 2.54 cm
1 mile = 1.609 km
1 statute mile=5280 feet (this is the normal kind of mile, it would be abbreviate mi, NOT m)
1 nautical mile= 6080 feet
1 nautical mile= 1.853 km
1 m = 100 cm
1 km = 1000 m
–6
1 µ = 10 m ( the symbol µ , like m in the Greek alphabet, stands for micron unit)
-9
1 nm = 10 m
–8
1 Å = 10 cm=10-10 m (the symbol Å stands for Ångstrom units)
8
1 AU = 1.496x10 km (AU stands for Astronomical Unit, the average distance from the Earth to the
Sun)
12
1 ly = 9.46x10 km (ly stands for light year, the distance that light would travel in a year)
1 pc = 3.26 ly
13
1 pc = 3.0857 x10 km
(pc stands for parsec, the distance at which parallax shift due to the observer moving by 1 AU is one
second of arc)
3
1 kpc = 10 pc
6
1 Mpc = 10 pc
Triangles
If you know two sides and one angle,
you can finish drawing the triangle.
Which means, the triangle is known.
e.g.
If you know all three sides, you can also finish the triangle, i.e. draw
and stick it together in only one way.
So you know the entire triangle
On the other hand, if you know all the angles, you know the shape of
the triangle, but not its size. Triangles with the same shape as one
another are called similar, like the following three triangles.
Chapter I Getting Your Bearings, The Sizes of Things
8
The angles in any plane triangle always add up to 180 degrees.
Angles
o
o
360 in a circle, 180 in the angles of a triangle or on one side of a straight line
o
1 =1 degree=60' ( minutes of arc)
1'=60" (seconds of arc)
Areas
Triangle Area = 1/2 Base x Height
Rectangle Area = Base x Height
Circle
Area = π Radius 2, where π = 3.14159...
The circumference of a circle is Circumference = 2 π Radius. The number of degrees in a circle is 360.
Trigonometry, just so you know
A special, useful class of triangles is the right triangle. Since one of the angles in it is always 90
degrees, the right angle, and there are 180 degrees total, there are 90 degrees left among the other two
angles. Once one of the angles is known, the other is simply the remainder of the 180 degrees.
Since the shape of a right triangle is known once one of the smaller angles is known, we can
understand the relations between the lengths of the sides. The relations are tabulated as the functions
shown above. The ratios of the lengths of the sides can be found in books or on scientific calculators.
Be sure that the calculator is in degrees mode to use it to find a trigonometric function.
o
One more quick trick. If an angle is smaller than about 10-15 , then sine or tangent of the angle is about
the same as the size of the angle in radians.
Angles
o
o
360 in a circle, 180 in the angles of a triangle or on one side of a straight line
o
1 =1 degree=60' ( minutes of arc)
1'=60" (seconds of arc)
o
1 radian = 57.295...
o
2 π radians =360
Areas
Triangle Area = 1/2 Base x Height
Rectangle Area = Base x Height
Circle
Area = π Radius 2, where π = 3.14159...
Sphere
Area = 4π Radius 2
The circumference of a circle is Circumference = 2 π Radius.
Chapter I Getting Your Bearings, The Sizes of Things
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