1. What are angles?
Last time, we looked at how the Greeks interpreted measurment of lengths.
However, as fascinated as they were with geometry, there was a shape that was
much more enticing than any other : the circle. The circle is a very easy, natural
object to picutre in your head. Actually producing a perfect circle is much more
difficult. Ancient cultures wanted to know all they could about this glorious shape,
its size, its counterparts, it boundary.
In order to look at the circle more carefully, we must first learn a bit about
angles. Imagine you have two rays or lines with a common endpoint, as pictured
below in figure 1. The angle is the amount of rotation separating these two rays.
In the figure below, as is customary, we denote the amount as θ. “Angle” comes
from the Latin angulus, meaning “corner”. Greek has its own cognate, αγκύλoς,
meaning “crooked”. Of course, we have not yet decided on a unit of measurement
for angles, so it does not make much sense to talk about measuring them yet.
θ
Figure 1
2. A brief history of angles.
As far back as Babylon, ancient cultures have been studying angles. We know
that the Babylonians split the circle into 360 units of measurement. We still use
this number 360, and we know them as degrees. We know that an angle of 180◦
creates a flat line. Or an angle of 0◦ means that the two rays overlap. The degrees
repeat every 360◦ . That is 0◦ is the same thing as 360◦ , and 90◦ is the same thing
as 450◦ , which is the same as −270◦ , and so on.
How did the Babylonians stumble across 360 as a unit of measurement, which
we still use today? The answer is unknown, but some recently-uncovered ancient
Babylonian tablets give us some clues. Some speculate that since an Earth years
as about 360 days, the Babylonians thought 360 would be a fit number to describe
a circle. A much more likely speculation is observing their connection between the
hexagon and their number system. Consider figure 2 below.
1
2
The Babylonians knew that the perimeter of a regular hexagon was equal to the
radius of a circumscribed circle. We also know that the Babylonians used a base
60 number system.
Figure 2
So it is very likely that the Babylonians attributed 60 units to each side of the
hexagon. Its resulting perimeter would then be 360!
As a quick aside, if you have never been exposed to number systems that use
something other than base 10, it is good to experience them. We use the Arabic
number system with ten different symbols : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. To represent
numbers beyond 9, we have to start putting symbols in the ”tens” place or ”hundreds” place, etc. The Babylonians had sixty different symbols in their number
system. For example, suppose that they used the symbol δ was their symbol for
our 59. Then our 10 would be their “60” (1 in the sixtys place, 0 in the ones
place). 11 would be their “61”. 1δ would be their 119 (1 in the sixtys place, 59 in
the ones place). It is a little strange to get used to, but that is how their culture
counted.
3. Back to the Greeks
While the Greeks were extremely fascinated by circles and arcs, they used a tool
or concept that made approximating their measurements much more easily. They
used something called a chord. Refer to figure 3 below. Imagine we take a piece
of the circle, so some arc. This arc will have an corresponding angle θ. In figure
3, we can see two rays extending from the center of the circle to edge. θ is the
angle between them. The cord is the straight line between the two points where
the rays meet the edge of the circle. (The chord is drawn in red in figure 3.)
3
θ
Figure 3
Notice that the two rays and the chord make an isosceles triangle. Also note
that, if we know what θ and the radius of the circle are, we also know all the
lengths of the triangle as well as the three angles. The Greeks were much more
familiar with triangles than arcs, so the chords proved to be a useful tool.
4. Radians
We are also familiar with another unit of measurement for angles : the radian.
Refer to figure 1.8 in your text; r is the radius of a circle, θ is an angle, and s is
the length of the arc determined by θ. The radian is defined as the ratio of the
s
s
length of the arc to the radius, or . So the angle in radians will be θ = . A
r
r
natural question to ask about the radian is if it depends at all on the size of the
0
circle we are using. i.e. if you have θ = rs = rs0 = θ0 , are the angles represented by
θ and θ0 actually the same angle? We would like to show that this definition does
not depend on the circle we are using.
Refer to figure 1.9 in your text. Imagine we have two circles, one with radius r
and another with radius R. Let’s assume R > r. We will pick an arbitrary angle
θ. Let s be the corresponding arc length determined by θ on the smaller circle.
Let S be the corresponding arc length determined by θ on the larger circle. We
would like to show that rs = RS . To prove this, we will use the ancient greek tool,
chords.
We will now partition the wedge into pieces. The number of pieces doesn’t
matter. In figure 1.10 in your book, the wedge is partition into 4 equal pieces. We
then draw the chords for each corresponding piece for both circles. In the figure,
d4 represents the chord in one of the pieces on the smaller circle, and D4 represents
the chord in one of the pieces on the larger circle. Remember that chords create
triangles that we are able to work with more easily. It is important to note that
each of the triangles inscribed in the smaller circle is similar to those inscribed
in the larger one. This is because the angles are equal. Also note that dr4 = Dr4 .
4
There was nothing special about 4, we could have chosen any other number n. In
general, we would get that drn = DRn .
Now try to think about what happens as that number n get very large, say
n = 1, 000, 000. This means we would partition the wedge into 1, 000, 000 pieces.
More importantly, the sum of all the dn get closer to s as n gets larger. So for
very large n, ndn ≈ s. (Also, nDn ≈ S.) This means that ndr n ≈ rs for very large
n. We write
s
nDn
S
ndn
=
and lim
= .
lim
n→∞ R
n→∞ r
r
R
Here, the lim means that we are pushing n arbitrarily closer to ∞, or rather
n→∞
making n very large.
Finally, since
dn
r
=
Dn
R
for any n, we get
ndn
nDn
S
s
= lim
= lim
= .
n→∞ R
r n→∞ r
R
s
S
So we have proved r = R . Thus, the notion of “radian” does not depend on the
size of the circle and is, in fact, a good unit for measurement of angles.
5. The length of a circle
Now let’s consider a circle of radius 1. If we let θ be 180◦ , the corresponding
wedge will be a semi-circle. In radians, our angle is the circumference1 of the semi-circle . We
give a special name to this ratio, and we call it π. Because the notion of radian
does not depend on the circle, we can say that for any circle, π is the ratio of the
circumference of the circle to its diameter. (This is the same as before, but we
multiplied the ratio on top and bottom by 2.) So for the circle of radius 1, the
circle has circumference 2π. For any circle, the circumference is then c = 2πr.
One final note is that 180◦ = π, and thus the sum of the angles of a triangle is π
radians. Refer to table 1.1 in your text for other values.
6. Estimating π
Although the Greeks did not use radians, they had noticed this universal constant of the ratio between a circle’s circumference and its diameter. Even earlier
cultures had happened upon it. There are references in the Bible to estimations of
π. King Solomon gives plans for a pond with diameter 10 cubits and circumference
30 cubits. Obviously he, as well as many other ancient cultures approximated π
by 3. The Greeks were able to use more advanced techniques to estimate π, and
were actually able to estimate it to a few decimal places. Below we give a proof
that 3 < π < 3.47.
5
Refer to figure 1.13 in your text for this proof. Consider a semi-circle with radius
1, label the center as C. We know a semicircle has circumference π. We wil split
this semi-circle into three equal parts as shown. Each part then has angle π/3.
Let A and B be the points where our cuts intersect the semi-circle as shown in the
figure. Now let AB be the chord between the points A and B. Since all three of
these angles are π/3 (why?), this triangle formed by the chord is equilateral. Split
this wedge (not the triangle) in half by drawing a line from C to the midpoint of
the arc, call it P . This is again shown in the figure.
Draw a line tangent to the circle at this point P and join this with extensions of
the lines through A and B forming a slightly larger triangle similar to our ACB
triangle. Call this meeting points A0 and B 0 . Once again, this is shown in the
figure. Again this triangle A0 CB 0 is equilateral (why?), so its angles are all π/3 as
well. Let x√be the length A0 P and P B 0 . Then the right trangle CP A0 have sides
1, x, and 2x. This is because of the Pythagorean theorem, 1 + x2 = (2x)2 , or
3x2 = 1, or x = √13 . So the length A0 B 0 has length 2x = √23 .
What does this mean? Imagine drawing a hexagon inside the circle, much like
in figure 2. We know this has perimeter 6, and we can see that the circumference
of a circle of radius 1 is larger than 6. So π > 3. Now imagine we draw a hexagon
containing a circle. What we have essentially just shown is that its perimieter will
be √123 . We can also see that this is larger than the circumference of the circle. So
√
we have π < √63 = 2 3 ≈ 3.47. So we have estimated π between 3 and 3.47.
This technique is similar to the one Archimedes used in his estimations of π. He
used regular polygons with many more sides. He used 96-sided polygon! Notice
that as regular polygons increase in number of sides, they appear to become more
and more like circles. Archimedes noticed this phenomenon and used 96 sides. He
then drew a 96-gon inside a circle or radius 1 and one outside. Then came the
tedious task of determining both of their perimeters. He knew that π would be
in between these two numbers. So he was able to show that 3 10
< π < 3 17 , or
71
3.1408 < π < 3.1429. This is pretty darn good for over 2000 years ago!
One final thought. π is a tricky number to calculate. The Greeks strived to find
it as they believed it should be a “number.” Unfortunately, it turns out that it is
not a “number” (in the Greek sense), because it is irrational. It has no predictable
decimal expansion and certainly cannot be written as a fraction of integers. One
of the most important numbers in existence turns out to be something relatively
foreign. The mysteries of the Universe!