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.

Go to Top

Main Menu