What is a circle?
By Murray Bourne, 11 Apr 2011
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.
11 Apr 2011 at 7:43 pm [Comment permalink]
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:
14 Apr 2011 at 12:19 am [Comment permalink]
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
23 Apr 2011 at 11:46 am [Comment permalink]
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
5 Jun 2011 at 1:20 pm [Comment permalink]
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.
4 Feb 2016 at 11:07 am [Comment permalink]
A circle is the locus of a point on which all points are equidistant from a fixed internal point known as the center.(sorry, for grammatical mistakes, if any)
18 Mar 2016 at 2:30 pm [Comment permalink]
Circle,1.any object which is perfectly round; ie. ⊙
2. Used in mathematics,social sciences, religion, astronomy, literature, and various forms of sports and arts , often deviating from the first definition .
3. Beginning at any point , going in 360° .
15 Feb 2018 at 1:09 am [Comment permalink]
Three ways to define a circle mathematically. 1)Locus of points:The set of points in a plane equidistant from a fixed point called the center.
2)Conic Sections:The intersection of a plane and a right circular cone such that the plane is perpendicular to the axis of the cone.
3)Algebraically:The set of ordered pairs satisfying the equation (x-h)^2+(y-k)^2=r^2 where (h,k) is the center and r is the radius.