Is a circle a continuous line or an infinite number of
angles?

Answer by Kermit Rose.

Viewing a circle as an infinite number of angles is unusual
but justifiable. A circle may viewed in any of several ways.

Ancient Greek mathematicians viewed a circle as a polygon
with an infinite number of sides. This is very close to the
idea of a circle as an infinite number of angles.

Draw a square. It certainly is not the same as a circle. Now
draw a hexagon. This is a polygon with six sides. It's still
not a circle, but its more like a circle than the square.

Now draw an eight sided polygon. One way to do this is to
start with the square. Find the midpoint of each side of the
square, and find the four points slightly outside the square
near those midpoints that when combined with the four corners
of the square, make the vertices of a 8 sided polygon.

In like manner, you can construct a 16 sided polygon, a 32
sided polygon, etc. It won't take long before what you've
drawn looks exactly like a circle.

This is why a circle may be thought of as a polygon with an
infinite number of sides.

How would we justify the idea of the circle as a continuous
line or an infinite number of angles?

You can draw a circle without lifting the pen from the paper.
This is what we mean when we say a circle is a continuous
curved line. A circle has the special property that it has a
center in its interior which is equidistance from all points
of the circle circumference.

The line segment connecting the center to a point of the
circle is called a radius. The plural of radius is radii.

On the other hand, it is very useful to associate each point
of the circle with an angle. How do we do this? First we
associate an angle with an arc of the circle. Draw two radii
from the center to the circle. The angle of the arc is
defined as equal to the angle between the radii. If you
choose the radii so that there is a right angle between them,
then the corresponding arc on the circle is 1/4 of the entire
circle.

We define the angle of an arc as its length divided by the
length of the radius.

The length of 1/4 of the circle circumference is 1/2 pi times
the radius. So the angle associated with 1/4 of the circle is
1/2 pi. A right angle is equal to 1/2 pi. This is why we say
90 degrees is 1/2 pi.

We use the word radians if we want to make clear we are
talking about angle measure. We then say 90 degrees is 1/2 pi
radians.

To associate each point of the circle, we make conventions as
follows. Draw a circle. Draw the horizontal diameter.
Extend the diameter to infinity in both directons

You have now drawn the x-axis. Now draw the vertical
diameter, and extend it to infinity in both vertical
directions. You have now drawn the y-axis.

Now locate the point of the circle on the x-axis to the right
of the center of the circle. This point is to be associated
with the angle of zero degrees.

Now move counterclockwise along the circle from the zero
degrees point. The point at an arc distance equal to the
radius of the circle is to be associated with the angle of
one radian.

If you continue counterclockwise exactly halfway around the
circle from the zero point, you will have traversed the
distance of pi times the radius of the circle. This follows
from the formula for the circumference of the circle being 2
* pi * radius.

The point halfway around the circle is to be associated with
the angle of pi radians. It is also 180 degrees. This is why
we define pi radians to be equal to 180 degrees.

All the other points of the circle are assigned an angle in
proportion to the counterclockwise arc distance from the zero
point.

Now that we have assigned each point an angle, it is clear
that these angles completely characterize the circle. Any
question we could answer about the circle in terms of it
being a continuous line could also be answered in terms of it
being all the angles between zero radians and 2 pi radians.

A circle may be thought of as

continuous line, or
a polygon with an infinite number of sides, or
an infinite number of angles.

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