What is a circle?

By Murray Bourne, 11 Apr 2011

japanese flag
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:

circle - curve

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

circle - closed region

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 4 Comments below.

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4 Comments on “What is a circle?”

  1. Dalcde says:

    I believe that a circle is a curve, mathematically. We talk about “equation of a circle” which defines a curve (the region enclosed would be described by x^2+y^2<r^2). The region inside the curve is a disk (or disc).

    In our everyday life, however, a circle can refer to both the curve and the region enclosed in the curve.

    Read Wikipedia for more information:

    In everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.

  2. Yan Kow Cheong says:

    Based on the “abnormally high” number of theorems on the circle, no doubt the circle is often taken to be the most beautiful curve in the all of mathematics – simple but not simplistic.

    There are enough properties of (and thousands of conjectures to work on) the circle to keep one mathematically busy for a three- or four-scores-and-ten lifespan.

    Let me end on a spiritual note:

    “The nature of God is a circle of which the center is everywhere and the circumference is nowhere.” Anonymous

    Circularly yours

  3. Mehdi says:

    The easiest way to define a circle is “The locus of points equidistant from a fixed point i.e. the center of the circle”…
    Also x^2+y^2=r^2 implies that only the boundary and not the region inside the circle

  4. MJ56003 says:

    But didn’t we all have classes that asked “What is the area of this circle?” and “What is the circumference of this circle?”

    If a circle is only a line, then it would have an area of zero.

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