Recall
the Euclidean postulate which states that two lines intersecting at a
single point uniquely determine
a plane. Moreover, these two lines divide the plane into four distinct
regions, referred to as angles.
Definition: An angle is a region in
the plane which is partially bounded on two sides by two lines
intersecting at a single point. Note by this definition the angle itself
consists of all the points in a particular
region of the plane. For example, in the Cartesian coordinate system (number
plane), the four quadrantal
angles are referred to as Quadrants I, II, III, and IV.
The lines forming the boundaries of an angle are called the sides
of the angle, and the point where the
lines intersect is called the vertex of the angle. Let
A be the point of intersection (the vertex), and let
B and C be points on the two sides of the angle. Then
the angle is referred to symbolically as (Angle)BAC.
It doesn't matter which of the two sides is listed first, so (Angle)CAB
is the same as (Angle)BAC.
That is, the order of the points labeling the angle does not matter, as
long as the vertex is listed in the middle.
Now, planes are infinite
in extent (this is one of Euclid's "axioms"), which may present
a problem in deciding
how angles should be measured. However, there is another approach using
circles which may be used to
define angles as being bounded in extent. Let a circle C
be given, centered at vertex A, and let P and Q
be
the points where the circle intersects the sides of the angle. Then (Angle)PAQ
(equivalently, (Angle)QAP)
is referenced within the circle as a central angle (since
the center of the circle is the vertex of the angle).
In this setting, the region of the plane bounded by the radii AP
and AQ and the arc PQ is called a sector
of the circle. (Note: an arc of a circle is
defined as a portion of its circumference lying between two end
points, just as a line segment is a portion of a line lying between two
end points.) Also in this setting, the
angle PAQ is said to be subtended by the arc
PQ. Note that if a different circle C'
is drawn centered at
A but with a different radius, then C'
intersects the sides of the angle at different points P' and
Q', but the
angle P'AQ' is supposed to be the same as the original angle
PAQ, which again raises the question of how
angles are supposed to be measured.
Angular measure
is a way to compare the relative "size" of two angles. Angular
regions of the plane are
of infinite extent, so it is impossible to use planar measure (area) to
define the size of an angle, and it is also
impossible to use circular arclength alone to define angular measure,
as discussed in the paragraph above.
Angular sizes are not measured in terms of an "absolute" unit
of angular measure, such as what is done for
length (the meter) or weight (the gram). Instead, angles are measured
relative to each other, or relative to
the angle representing the entire plane (or, for central angles, the entire
circle). It is perhaps easier to see
how central angles might be measured in terms of fractions of a circle,
but let us return to the original
definition of an angle and see how it might be defined in terms of fractions
of the plane.
Given two lines interesecting
at a single point, defining four planar angles, use the Euclidean postulate
of
measure so that both lines represent real number lines, intersecting at
the common origin (0,0). Each number
line is marked off in terms of a unit of distance, which is marked off
from (0,0) in both directions along both
the positive and negative directions of each number line, defining the
four "compass points" (1,0), (1,0),
(0,1), and (0,1). Consider now the following four distances between these
four points:
 D1 = distance
between (1,0) and (0,1) = the Quadrant I distance
 D2 = distance
between (1,0) and (0,1) = the Quadrant II distance
 D3 = distance
between (1,0) and (0,1) = the Quadrant III distance
 D4 = distance
between (1,0) and (0,1) = the Quadrant IV distance
If the number lines
intersect to form angles in such a way that all four of these distances
have the same
measure, then the lines are said to be perpendicular,
and all four quadrantal angles are defined to have
the same measure. In this case each angle is said to be a right
angle, and so a right angle has measure
representing onefourth of the plane when the plane is divided into four
equal parts. (Note: "perpendicular"
is based on a Latin phrase meaning "vertical" in the sense that
a weight hanging from a string will cause the
string to be vertical as opposed to the horizontal ground ("horizontal"
of course referring to the ground's
horizon), and if you want your building to stand up straight then the
walls must form the "right" angle with
respect to the ground, i.e. erected to follow the same direction as the
perpendicular line.)
If the two number
lines interesect in such a way that the four distances D1 through D4 are
not all the same,
then the angles formed by the lines will have different measure, either
greater than or less than the measure
of a right angle. However, since it is completely arbitrary along each
number line which direction is chosen
as positive and which direction is negative, it will always be true in
the list above that D1 is equal to D3 and
D2 is equal to D4. Thus, if relative distances are to be used in defining
angular measure, when two lines
intersect at a single point they must always form two pairs of angles
such that within each pair of angles
the two have the same measure. Thus, two lines interesecting at a single
point always form two pairs of
angles of the same measure, referred to as vertical angles,
and if the lines are not perpendicular then in
one pair of
vertical angles each angle has measure less than that of a right angle
(acute angles) and in the
other pair of vertical angles each angle has measure greater than that
of a right angle (obtuse angles).
Units of
Angular Measure
Now that the idea
of relative angular measure has been established, return to the concept
of central angles
defined in terms of a circle. By definition, right angles must divide
the circle into four equal parts, and so,
once it is decided what angular measure an entire circle has, a right
angle must have angular measure
equal to that of a quartercircle. There are two basic circular units
of angular measure:
Degree
Measure
defines the circle as having measure 360 (approximately the number of
days
in a year), and so in this system a right angle would have measure 90
degrees. This was the system
developed by early mathematicians, in part because 360 is divisible by
many whole numbers.
The second system is that of Radial Measure, which measures
the size of a central angle subtended
by a given arc in terms of the ratio of the arclength along the circumference
of the circle to that of the radius
of the given circle. Radial measure, the units of which are called radians,
must then be given in terms of the
number pi, which is defined to be the ratio of any circle's circumference
to its diameter (this ratio is a little
bit bigger than 3). That is, C = pi*D, where
C is the circumference of a circle and D is its diameter.
Since the radius (r) of a circle is twice its diameter, and since
radians measure the ratio of circumferential
arclength to the radius, the entire circle must have radial measure equal
to 2*pi radians. Thus, in radial
measure, a right angle (quartercircle) has measure pi/2 ("pihalves")
radians.
Note that both these
units of angular measure are defined by dividing a circle into a certain
number of
parts, with the entire circle measured as either 360 degrees or 2*pi
radians. Thus, one degree is equal
to pi/180 radians, and one radian is equal to 180/pi
degrees. Since pi is a little bigger than 3, one radian
is a little bit less than 60 degrees.
Other units of angular
measurement are based on dividing a right angle into a certain number
of equal parts.
For example, gradial measure divides a right angle into
100 decimal parts, so that 90 degrees is the same
as 100 grads. Decimal angular measure is useful in engineering calculations,
which is often applied to the
military, so there is also a finer subdivision of angular measure used
in military applications referred to as a
mil, with different definitions depending upon the application,
but generally if you bisect a grad and then
bisect it again you get a mil, which means there are 400 grads and 1600
mils in a circle.
The number of degrees in a circle, 360, is a multiple of 2, 3, 4, 5,
6, (not 7, a magic number), 8, 9, 10,
(not 11, the next magic number), 12, (not 13 or 14), 15, 20, 30, 45, 90,
and 180, so ancient mathematicans
found 360 to be useful in many calculations, especially in astronomy because
360 is very nearly the number
of days it takes the Earth to go around the Sun. Suppose life had developed
on Mars instead of Earth?
How many degrees would there be in a Martian circle, a Jovian circle,
or an Alpha Centaurin circle?
How would you define angular measure?
