What is a Perfect Circle?

Question

People often play around with the word "perfect circle" but I have always been curious as to what exactly is a "perfect circle". It is easy to imagine, but to me, even when given precise geometrical equipment, it seems impossible to verify that a circle is perfect unless you have infinite time. The way I understand a "perfect circle" is as follows:

It is a shape consisting of all the infinite points that are a distance $r$ from a central point.

If this interpretation is correct, however, it seems impossible to know whether a circle is actually perfect unless we measure the distance $r$ from the center to each of the infinite points. Is there, then, a better way to think about what exactly a "perfect circle" is and to be able to accurately verify if a circle is perfect without the need for infinite time?

Answer 1

We have a phrase, "to come full circle" which means "to get back to where one started." If there is a "full circle" then there are not-full circles. Otherwise "full circle" is redundant. If all circles are full, then there's no sense in saying "full circle."

We have changed the definition of "perfect" over the years. It used to mean "complete." When something was perfected, that meant you were done working on it. (So you can see why "perfect" has come to mean "flawless.")

When we do compass-and-straightedge geometry, we are always drawing little scratches of arcs, which are not complete circles. So a complete or perfect circle is a specific geometric figure. Incomplete circles don't bound a region in the plane, so there's no area to talk about. So one would say, "The area of a perfect circle is..." to indicate that the whole circle is intended.

Just re-read anything you've read before about "perfect circles" and change the word "perfect" to "complete." Things will make a lot more sense then.

Edit: Let me add that you should take note of the word "all" in your definition of a perfect circle. It's the "all" that makes it perfect.

Answer 2

A long comment

Your definition for a perfect circle is true, but a perfect circle is hard to come by especially in nature

It is a shape consisting of all the infinite points that are a distance $r$ from a central point.

The circles in geometry are perfect, but finding a perfect circle is difficult because we would have to measure all the distance from its center, so we need a perfect measurement to verify a perfect circle.... Another way to verify a perfect circle is to measure the number $\pi$, it is the ratio of a circles circumference to its diameter, but whenever we measure this we can only get few digits of $\pi$ in a prefect circle we are expected to get all the digits of $\pi$

Say in a right angle triangle, two sides are $1$ and $1$, the hypothenuse side would be $\sqrt{2}$, this seem like a perfect shape but in construction , the hypothenuse is $1.41$, If we agree the shape was perfect then the hypothenuse would be exactly $\sqrt{2}$

Infact any measurements that tells us that the hypothenuse is exactly $\sqrt{2}$ would contradict on the adjacent length, because it would also have infinite digits

This idea applies to all objects in geometry, let's say a circle with unit length was constructed, it's perimeter must $2\pi$, any measurements that proofs it exactly equal to $2\pi$ ( i mean having all it's digits ), would contradict on the length of the circle

Now I begin to ask, what is $0.9999999\dots$ and what is $1$, are they equal ?

So therefore nothing is perfect, perfects circles are not real, just an abstraction