Angles and Angular Measurement
Angles and Angular Units
An angle measures the difference between two directions just as a length describes the spatial distance
between locations. Just as various units (feet, meters, light-years) can be used to describe length or
distance, so different units can be used to measure angle. The mathematicians and philosophers of
classical times preferred to express angles as fractions of the full circle, with values lying between zero and
one. Today we usually use a system originating with the ancient Babylonians in which the full arc of the
circle is divided into 360 equal parts called
degrees
(°). The origin of this system is astronomical, one
degree (1°) is approximately equal to the daily motion of the Sun among the stars of the celestial sphere.
Mathematicians prefer to represent the full circle by 2π = 6.283185...
radians
where π (“pi”) is the ratio of a
circle’s circumference to its diameter*. Thus 1 radian is about equal to 57 degrees.
Angular Nomenclature
Two opposite direction (North and South, Zenith and Nadir, for examples) are separated in angle by a half
circle (θ =1/2), 180 degrees (θ = 180°), or π radians (θ = π). A pair of perpendicular directions (North and
East, Vertical and Horizontal, ..) are said to form a right angle corresponding to a quarter circle, 90°, or π/2
radians. Angles less than a right angle are said to be acute, those greater than 90° are obtuse. Notice that
two lines which make an angle θ with respect to one another also make an angle of 360° - θ with respect to
each other; it just depends upon the direction in which one chooses to measure things:
θ < 90° θ = 90° θ > 90°
. . . . .
An acute angle A right angle An obtuse angle
(about 40°) (exactly 90°) (about 130°)
Very small angles can be expressed as small fractions or as decimal parts of the basic unit. One can, for
example, speak of thousandths of a degree. Historically, however, the degree (1° = 1/360 th circle) was
subdivided into 60 parts called
minutes of arc
(‘) or simply
minutes
(1° = 60’, 1’ = 1°/60) and each minute
divided into 60
seconds of arc
(“) or
seconds
(1’ = 60”, 1”‘=1’/60). Thus
1 full circle = 360° = 21,600’ = 1,296,000”
A useful conversion number to remember for astronomical purposes is that 1 radian = 206,265”.
Measuring Angles: Some Examples
A
protractor
is a circular disk with graduated markings. It is useful for measuring angles on a drawing. A
surveyor’s
transit
or a mariner’s
sextant
is basically a precision protractor equipped with a telescope or
other sighting device for more accurate measurements. The human body is the basis for some units of
length measurement (
e.g.
, the yard) and can also be used to measure angles in an approximate way: The
span of your open hand at arm’s length is roughly 20° and the width of your palm about 8° whereas the
width of your thumb subtends about 2.5°. It is instructive to “calibrate” your body parts against something
of known angular size: How many Moon diameters are covered by your big toe? How many “hand widths”
does it take to circle the 360° horizon. Note that, on the sky, the angular diameter of the Sun or Moon is
close to 1/2° and both appear to move about 15° per hour because of the Earth’s rotation. The “bowl” of
the Big Dipper is about 6° across.
*There are more obscure units: The traditional mariner’s compass divides the circle into 32 named
points
of the
compass. (Examples: north, south-southwest, north by northwest,..). Artillery men divide the circle into 6400
mils
for
purposes of aiming their guns. Just to add to the confusion, there is an “infantry mil” which is equal to 1.018 “artillery
mils” . Neither, of course, should be confused with the length unit (1 mil = 0.001 inch) nor the monetary unit (1 mil =
0.001 Israeli pound) nor with the abbreviation for milliliter which is a liquid volume of about one cubic centimeter......
