How we see circles
Viewing a circle as an infinite number of angles is unusual but justifiable. A circle may be viewed in 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 it’s 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 an 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 directions
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.