Discusses the concept of measurement in general before exploring how we might define a unit of measure for angles, and design a tool for the purpose (which, in its refined version, is what we now call a "protractor"). Shows how to use a protractor to draw angles of various sizes, then ends by introducing the concept of the radian measurement.

1. Measuring,
constructing,
and
using angles

2. Contents
Why Measure Angles?
How might we measure angles?
What is an “angle”?
What does “measurement” mean?
Our “measuring tool” for angles.

3. Contents (continued)
Measuring an angle
Drawing an angle of a given size
The “radian” unit of measure for
angles.

4. Why measure angles?

5. Why measure angles?
So we can figure out
how big the Earth is!

6. If we choose two places that are
north and south of each other,
A
B

7. then measure the distance
between them,
A
B

8. and also the highest angle the Sun
reaches at each place on a given day,
Highest angle at Location A Highest angle at Location B
Sky simulations made with the free astronomy program Stellarium
(http://stellarium.org/)

9. we can calculate
the size of the Earth.
The first person to do this
calculation was Eratosthenes, a
librarian at the Great Library of
Alexandria, Egypt, around 200 BC.

10. Actually, almost all of astronomy
and geography depends upon
being able to measure angles.
So do many jobs, such as
surveying.

11. So, how might we measure
angles?
First, let’s review what we mean by
“angle”, and “measurement”.

12. One definition of “angle” is
“A pair of rays that
have the same endpoint”.
The endpoint,
V, is called
the vertex.

13. You’ve probably learned that
angles are related to “turns”, and
that the same set of rays can be
made by two different turns.
For example, here are …

14. the two “turns” that take us from ray
VA to ray VB.
Counterclockwise turn Clockwise turn

15. Now that we’ve reviewed what
“angle” means, what do we
mean by “measurement”?

16. A good definition is found in The
Archimedes Codex: How a
Medieval Prayer Book is Revealing
the True Genius of Antiquity’s
Greatest Scientist:*
* Reviel Netz and William Noel, Da Capo Press,
2009, p. 41.

17. “To measure is to find a
measuring tool and apply it
successively to the object being
measured. Suppose we want to
measure a straight line.
According to the authors,

18. “For instance, suppose we want to
measure your height, which is
really saying that we want to
measure the straight line from the
floor to the top of your head.

19. “Then what we do is take a line
the length of an inch [this is our
measuring tool] and apply it
successively, well over sixty times,
but probably fewer than eighty
times to measure your height.

20. “Since this is very tiresome, we
have pre-marked measuring tapes
that save us the trouble of actually
applying the [one-inch line]
successively,

21. “but, at the conceptual level,
successive application is precisely
what takes place.”

22. That definition of “measurement”
needs some explanation. For
example, when we read
“To measure is to find a measuring
tool and apply it successively to the
object being measured,”

23. we probably thought of “measuring
tools” as rulers, etc.
Actually, the authors meant
something quite different.

24. They meant that to measure a
length (that’s the example they
give), the “tool” we choose is a line
segment with a convenient length.

25. The authors mentioned “an inch” as
an example of a convenient length,
but we could use a segment of any
length we like.

26. For example, we could use this
yellow segment:

27. Now, suppose that we wanted to
“measure” this red segment.

28. We’d start at one end of the red
segment, and “apply” the yellow
segment to it repeatedly, until we
got to the other end …

29. 1

30. 2

31. 3

32. Here’s the summary of what we just
did to “measure” the red segment.
1
2
3

33. Because we had to apply the
yellow segment three times,
1
2
3

34. we say that the length of the red
segment is “3 yellow segments”.
1
2
3

35. For example, if our yellow segment
were an inch, the length of the red
one would be “three inches”.
1
2
3

36. As the authors of The Archimedes
Codex told us, this process would
be tiresome if we had do many
measurements.

37. For that reason, we make pre-
marked rulers and measuring
tapes,
1 2 3 4 5

38. so that we may “apply” several
“yellows” at once.
1 2 3 4 5

39. If we wished, we could also mark
off fractional parts of our yellow
segment, just as we do with inches
on a ruler.

40. Now that we’ve reviewed what
“angle” and “measuring” mean,

41. let’s learn the common ways
of measuring an angle.

42. As we’ve seen, the “measuring
tool” that we use for lengths is
some convenient segment.

43. However, when we measure
angles, the “measuring tool” that
we use is some sector of a circle.

44. For example, this one:

45. To measure an angle,

46. we align one side of our sector with
one of the angle’s rays,
Note: “Point” of
the sector must be
aligned with vertex
of the angle.

47. Note: “Point” of
the sector must be
aligned with vertex
of the angle.
then keep “applying” our sector
until we reach the other ray.

48. Note: “Point” of
the sector must be
aligned with vertex
of the angle.
then keep “applying” our sector
until we reach the other ray.

49. We use the same method for the
other direction of “turn”:

50. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

51. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

52. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

53. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

54. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

55. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

56. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

57. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

58. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

59. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

60. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

61. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

62. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

63. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

64. Note: “Point” of
the sector must be
aligned with vertex
of the angle.

65. Although we could choose any
sector as our “measuring tool”, the
most-common is the one we get
when we divide a circle into 360
equal sectors:

67. The angle enclosed by each of
those sectors is called one degree.
We abbreviate that as “ 1° ”.

68. Important:
No matter how big our circle,

69. a “ 1° ” sector always contains the
same angle.

70. So, now we have our 1° measuring
tool.

71. Let’s use it to measure an angle.

82. We had to apply our “ 1° ” tool
eleven times,

83. so we say that the angle “measures
11°”.

84. Now that we have our angle-
measurement tool, and know how
to use it, we can look for ways to
make a more-convenient version of
it.

85. We already know that we can make
a more-convenient version of a
length-measurement tool by pre-
marking “something” at chosen
intervals, to make a tape or ruler.

86. For measuring angles, we can pre-
mark 1° sectors on something:

87. However, this tool is “messy”, so
we “clean it up” by …

88. leaving marks only at the center
and edges of our circle.

89. Those are all we need for
measuring an angle.

90. To make a “nicer” tool, we can
number the degrees, and add a few
other extras, to make the tool we
call …

91. a protractor …

92. which comes in several versions.

93. To measure an angle with a
protractor,

94. we align a “zero” line of the
protractor with a ray of the angle …

95. we align a “zero” line of the
protractor with a ray of the angle …
Note where
protractor
aligns with
vertex.

96. we align a “zero” line of the
protractor with a ray of the angle …

97. we align a “zero” line of the
protractor with a ray of the angle …

98. we align a “zero” line of the
protractor with a ray of the angle …
Note where
protractor
aligns with
vertex.

99. then read how many degrees we’d
“apply” to get to the other ray.
90°

100. Notice that we’ve measured the
shorter of this angle’s two “turns”.
90°

101. Here’s the longer “turn” for this
angle.

102. How many degrees should it
measure?

103. We’ll discuss that later.

104. The 90° angle (also called a right
angle) is an important one to know.

105. So is the angle that measures
180°. It’s called a “straight angle”.

106. So is the angle that measures
180°. It’s called a “straight angle”.

107. So is the angle that measures
180°. It’s called a “straight angle”.

108. So is the angle that measures
180°. It’s called a “straight angle”.

109. 90° 180°
Knowing these two angles…

110. helps us when we have to measure
an angle like this one.

111. Does it measure 60°, or 120°?

112. By counting, we can see that there
are 60 degrees between the rays.
60°

113. Also, 60° minus 0° is 60°. (That’s
what we read on the inner scale).
60°

114. 180° - 120° is 60°, too. (That’s what
we read on the outer scale).
60°

115. Our angle 90°
Finally, our angle (the one we want
to measure) is smaller than 90°.

116. All of these observations show that
our angle measures 60°, not 120°.
60°

117. For comparison, here’s an angle
that does measure 120°.

118. For comparison, here’s an angle
that does measure 120°.
120°

119. Our angle 90°
We can see that it’s bigger than
90°,

120. Our angle 180°
but smaller than 180°.

121. Please remember that when we
say that these angles measure 60°
and 120°,

122. we’re talking about the measure of
the smaller of their “turns”,

123. not the larger one.

124. These larger “turns” measure more
than 180°, but less than the 360°
that there are in a full circle.

125. They’re examples of what are
called “reflex angles”.

126. So is the “larger turn” of a 90°
angle, which we saw earlier.

127. We’ll now look, briefly, at how to
find the measure of a reflex angle.

128. First, we can break the reflex angle
into a straight angle (pink turn), …

129. and a 60° angle (orange turn).

130. The measure of the reflex angle is
the sum of these: 180° + 60° = 240°.
60°
180°

131. 240°

132. A second way is to remember that
in a full circle, there are 360°.

133. Therefore, the number of degrees
in the reflex angle has to be …

134. 360° minus 120° (= 240°).
360°
120°

135. 240°

136. A third way to measure a reflex
angle …

137. is with a full-circle protractor.
240°

138. Here are links to more information
about reflex angles:
Reflex angles
http://www.mathsisfun.com/reflex.html
Re-entrant angles
http://www.thefreedictionary.com/re-
entrant+angle

139. Besides wanting to measure
angles, there are times when we
want to (or must!) draw them.

140. Someone will tell us (or we will
choose) the following things:

141. Where the vertex is,

142. the ray from which the angle is to
start,

143. the direction of the “turn”, and

144. the measure of the angle that we
are to draw.
105°

145. Now, just as we did for measuring
angles, …
105°

146. we align a protractor with the vertex
and ray.
105°

147. Next, we go in the direction of the
given turn,
105°

148. for the required number of degrees.
105°
Here’s the
mark for 105°.

149. We make a little dot there on our
paper, …
105°
Here’s the
mark for 105°.

150. then remove our protractor, ...
105°

151. and draw a ray from the vertex
through our dot.
105°

152. We’ve now constructed the
required angle.
105°

153. We’ll finish this discussion of
angles by mentioning another
common “tool” for measuring them.

154. We imagine making that tool by
drawing an circle, and one of its
radii, …

155. then making a segment as long as
the radius, …

156. sticking one end of the segment to
the circle, …

157. and bending the segment to fit the
circle.

158. Now, we draw rays from the center
of the circle through the endpoints
of the segment.

159. The angle we’ve “cut out” of the
circle is defined as “1 radian”.

160. This “measuring tool” has a
property that makes it useful in
subjects that you’ll study later.

161. That property is that
In a full circle,
there are 2π radians.

162. Can you see why this is so?
Hint: What is the relationship
between the length of a circle’s
radius, and the length of its
circumference?

163. Summary
Rather than review the details of
how to measure angles, I’d like to
emphasize that …

164. Summary (cont’d)
• Many types of work depend
upon being able to measure
angles. For example, astronomy
and surveying.

165. Summary (cont’d)
• There are different ways of
measuring angles.

166. Summary (cont’d)
• When we need to find a way to
measure something, we can
invent our own “tools”.

167. Summary (cont’d)
• It’s often helpful to imagine some
simple version of a tool, or of
solving any problem, then look
for a way of making that simple
tool or technique more
convenient.

168. END