What is a circle?

Japanese flag

Most people would describe the Japanese flag as being "a red circle on a white background". But is it really, mathematically speaking?

Reader Irshad Hussain recently asked for "a clear definition of a circle." He wondered if the circle is only a boundry or does it include the whole interior also?

When you think "circle", do you see a curve, like this:

Or do you think of it as a region, like this?

Math Open Reference defines a circle as:

A line forming a closed loop, every point on which is a fixed distance from a center point.

This is the first diagram above.

The American Heritage Science Dictionary gives the following definition, also considering the circle as a curve, not a region:

A closed curve whose points are all on the same plane and at the same distance from a fixed point (the center).

Wolfram|Alpha also defines it as a plane curve. (And that's all. Even though it lists several important equations for circles, no mention is made of the property of equidistance from a point).

Google's definitions cover both cases, but give precedence to the region definition (the second diagram):

1. A round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center)
2. The line enclosing such a figure

Here's a definition that gives a broader view:

Ellipse in which the two axes are of equal length.

One of the silliest definitions is from the The American Heritage Dictionary:

Circle: A planar region bounded by a circle.

How can an object be bounded by itself? One could argue the definition itself is circular. 🙂

Is the circular region a disk?

The simplest solution is to define a circle as a plane curve and a disk as a plane region, bounded by a circle. However, "disk" to me suggests a 3-dimensional object (a very flat cylinder).

What are your thoughts on how we should define a cirlce?

See the 7 Comments below.