Angles
megaConverter #17
INFORMATION
PAGE
Introduction and
Overview
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Units of common measurement vary widely, from
time to time, and place to place. Most units of
measurement started as a reference to a physical
object or concept. The foot was the length of a
man's foot, the inch was the width of a man's
thumb, a furlong was the length of a plowed
furrow in a field, an acre was the amount of land
a man and two oxen could plow in a day, etc. At
first, most measurements were only
approximations, but eventually many country's
governments set each at a specific standard to
make commerce possible and fair. Often, when
people settled new lands, they used the names of
old measurements but set their own standards.
Other times, similar sounding measurement names
in different countries had greatly different
values. Some measurements were derived from other
types of measurements, such as a barrel weight
being the weight of a barrel of flour. Often, the
same measurement had different values depending
on the material being measured, such as a wine
tun and a beer tun, or a hank of wool and a hank
of cotton. These differences made sense to the
people that used them, but they seem odd today.
A Frenchman first defined what he called the
'meter' as one ten millionth of the distance from
the North Pole to the equator along the Prime
Meridian. It was later defined, in a more precise
method, as so many wavelengths of a certain color
of light. A liter was originally defined as a
cubic decimeter, and a kilogram was defined as a
liter of pure water at a specified temperature.
Later, the standard was changed such that a
kilogram mass became the standard and the liter
was derived as the volume of a kilogram of water.
This has caused the liter to become slightly more
than a cubic decimeter.
The International System (SI) was first
proposed in France in the 17th century, but was
not adopted by France until 1795. The system
defined that there was only one standard in each
measurement type and each unit greater or less
was a power of ten. This made conversions between
units much simpler. During the 19th century,
several countries made this system their
standard, but notably not Britain or the US. In
1965 Britain began changing to the metric system
as a condition of membership in the European
Common Market. The US government, recognizing the
problems of international trade, officially made
the metric system its standard in 1975.
Still today, units of common measurements
(nonmetric) are used throughout the world. It
would be hard to forget the foot, yard, mile,
quart, gallon, or acre because so many physical
objects were based on them. And for convenience
sake, it will always be easier to say "a
cup" than "two deciliters." It is
easier to envision a mile than a kilometer
because fence rows, city blocks, and farmland
measurements were originally based on the mile.
* Much of our written history still refers to
things in common units. The Bible does not refer
to meters or kilograms, but to cubits and stadia,
or shekels and drachma. Wouldn't it be nice to
know what they were talking about way back then?
Now you can use megaConverter! For a more
complete listing of ancient, foreign, and
obsolete measures, download our
'megaSpreadsheet' of conversions in MS Excel
format.
Glossary
of Conversions:
radian
When two straight lines meet, we say
the relative position of the two lines to one
another can be expressed as the angle between
them. If we hold one line fixed and rotate the
other about the meeting point, or vertex, the
moving line can rotate through a full circle
until it is back at its original position. At one
point in the rotation the moving line and the
fixed line will be coincident, one on top the
other. This point of coincidence is defined as
the zero point. Now imagine a circle centered on
the vertex of the two lines. The line segments of
both lines from the vertex to the circle can
represent the radius of the circle. Ancient
mathematicians determined that the circumference
of a circle could be determined by multiplying
the diameter of the circle by a constant that we
now call pi. This constant is an irrational
number, meaning that it cannot be represented
exactly by any integer divided by any other
integer. Pi is approximately 3.1416. This is
close enough for most uses. Mathematicians say
that a full circle is represented by an angle of
2*pi radians. If you think about it, this means
that a radian is the same angle you would have if
you took a circle’s radius and bent it
around the circumference of the circle.
gradient
There is one other angle measure used,
but only rarely, which is gradient measure. The
full circle is divided into 400 equal units. This
is handy only because the right angle (90
degrees) is now 100 gradients. Each individual
gradient can be considered 1 percent of a right
angle. This made calculations of things like
cannon shell trajectories slightly easier,
although with modern electronic computers, this
is no longer necessary.
degree
Radian measure is very important for
making calculations in geometry, trigonometry,
and calculus, but is rather burdensome to use
simply describing angles. Ancient Sumerians
devised a number system based on 60. They divided
the circle into 360 equal parts to describe the
angles of lines. This is useful because 360 is
divisible by many factors. We call each unit part
a degree. A full circle has 360 degrees. If we
pass 0 during our rotation around the circle we
keep going. Thus, 400 degrees is the same as
400360=40 degrees. In this way, we can speak not
only of angles but of rotations. 720 degrees is 2
full circles. Degrees were also broken down into
smaller units of minutes and seconds, where a
degree was 60 minutes, and a minute was 60
seconds.
point
Compasses use points which are 1/32 of
a circle or 11.25 degrees.
Note: Because of roundoff
errors, converting from very large units to very
small units or viceversa may not be accurate (or
practical). Conversion factors can be found by
converting a quantity of 1 unit to another unit
several steps above or below the first. You may
need to string several conversion factors
together to find the factor from a very large
unit to a very small unit, and then you can use a
calculator with sufficient digits to find your
answer.
mil
Military artillery trajectory calculations sometimes
use mils which are 1/6400th of a circle.
