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

Hello Charlotte!

You asked the question: Is a circle a continuous line or an infinite number of angles?

You might have expected me to chose one of these answers as the better way to look at the circle. This would not be the way mathematicians look at things.

The correct answer is that the circle is both a continuous line and the circle is also an infinite number of angles.

A circle is usually defined as a figure all of whose points are equidistance from another point called the center of the circle.

If we think of geometric figures as continuous lines or curves, then it is natural to think of a circle as a continuous closed curve around a center. A circle would then have the special property that it has a center in its interior which is equidistance from all points of the circle.

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 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 is 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 chose one point of the circle to be zero degrees, or zero radians. Then a point of the circle 1 radian from the zero point is assigned the angle of 1 radian. The point exactly halfway around the circle from the zero point is assigned the angle pi radians.

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.

That is why the correct answer is that a circle is both a continuous line and also an infinite number of angles.