A circle is the locus of points which are equidistant from some fixed point. It is in the plane is determined by a center and a radius. The center is a point in the plane, and the length of the radius is a positive real number, the radius being a line segmentMathworldPlanetmath from the center to the circumferenceMathworldPlanetmath. The circle consists of all points whose distanceMathworldPlanetmath from the center equals the radius.

Another way of defining the circle is thus: Given A and P as two points and O as another point, the circle with center O is the locus of points X with OX congruentMathworldPlanetmathPlanetmath to AP. (Hilbert, 1927)

Notice there is no definition of distance needed to make that definition and so it works in many geometriesMathworldPlanetmath, even ones with no distance function. Hilbert uses it in his Foundations of Geometry book Also used by Forder in his Foundations of Euclidean geometry book, c 1927

In this entry, we only work with the standard Euclidean norm in the plane. A circle has one center only.

A circle determines a closed curve in the plane, and this curve is called the perimeter or circumference of the circle. If the radius of a circle is r, then the length of the perimeter is 2πr. Also, the area of the circle is πr2. More precisely, the interior of the perimeter has area πr2. The diameterMathworldPlanetmathPlanetmath of a circle is defined as d=2r.

The circle is a special case of an ellipsePlanetmathPlanetmath. Also, in three dimensionsPlanetmathPlanetmath, the analogous geometric object to a circle is a sphere.


Let us next derive an analytic equation for a circle in Cartesian coordinatesMathworldPlanetmath (x,y). If the circle has center (a,b) and radius r>0, we obtain the following condition for the points of the sphere,

(x-a)2+(y-b)2=r2. (1)

In other words, the circle is the set of all points (x,y) that satisfy the above equation. In the special case that a=b=0, the equation is simply x2+y2=r2. The unit circle is the circle x2+y2=1.

It is clear that equation 1 can always be reduced to the form

x2+y2+Dx+Ey+F=0, (2)

where D,E,F are real numbers. Conversely, suppose that we are given an equation of the above form where D,E,F are arbitrary real numbers. Next we derive conditions for these constants, so that equation (2) determines a circle [2]. Completing the squares yields




There are three cases:

  1. 1.

    If D2-4F+E2>0, then equation (2) determines a circle with center (-D2,-E2) and radius 12D2-4F+E2.

  2. 2.

    If D2-4F+E2=0, then equation (2) determines the point (-D2,-E2).

  3. 3.

    If D2-4F+E2<0, then equation (2) has no (real) solution in the (x,y) - plane.

2 The circle in polar coordinates

Using polar coordinates for the plane, we can parameterize the circle. Consider the circle with center (a,b) and radius r>0 in the plane 2. It is then natural to introduce polar coordinates (ρ,ϕ) for 2{(a,b)} by

x(ρ,ϕ) =a+ρcosϕ,
y(ρ,ϕ) =b+ρsinϕ,

with ρ>0 and ϕ[0,2π). Since we wish to parameterize the circle, the point (a,b) does not pose a problem; it is not part of the circle. Plugging these expressions for x,y into equation (1) yields the condition ρ=r. The given circle is thus parameterization by ϕ(a+ρcosϕ,b+ρsinϕ), ϕ[0,2π). It follows that a circle is a closed curve in the plane.

3 Three point formula for the circle

Suppose we are given three points on a circle, say (x1,y1), (x2,y2), (x3,y3). We next derive expressions for the parameters D,E,F in terms of these points. We also derive equation (3), which gives an equation for a circle in terms of a determinantDlmfMathworldPlanetmath.

First, from equation (2), we have

x12+y12+Dx1+Ey1+F =0,
x22+y22+Dx2+Ey2+F =0,
x32+y32+Dx3+Ey3+F =0.

These equations form a linear set of equations for D,E,F, i.e.,


Let us denote the matrix on the left hand side by Λ. Also, let us assume that detΛ0. Then, using Cramer’s rule, we obtain

D =-1detΛdet(x12+y12y11x22+y22y21x32+y32y31),
E =-1detΛdet(x1x12+y121x2x22+y221x3x32+y321),
F =-1detΛdet(x1y1x12+y12x2y2x22+y22x3y3x32+y32).

These equations give the parameters D,E,F as functionsMathworldPlanetmath of the three given points. Substituting these equations into equation (2) yields

(x2+y2)det(x1y11x2y21x3y31) -xdet(x12+y12y11x22+y22y21x32+y32y31)

Using the cofactor expansion, we can now write the equation for the circle passing through (x1,y1),(x2,y2),(x3,y3) as [3, 4]

det(x2+y2xy1x12+y12x1y11x22+y22x2y21x32+y32x3y31)=0. (3)


  • 1 D. Hilbert, Foundations of Geometry Chicago: The Open Court Publishing Co. (1921): 163
  • 2 J. H. Kindle, Schaum’s Outline Series: Theory and problems of plane of Solid Analytic GeometryMathworldPlanetmath, Schaum Publishing Co., 1950.
  • 3 E. Weisstein, Eric W. Weisstein’s world of mathematics, on the circle.
  • 4 L. Råde, B. Westergren, Mathematics Handbook for Science and Engineering, Studentlitteratur, 1995.
Title circle
Canonical name Circle
Date of creation 2013-03-22 13:36:23
Last modified on 2013-03-22 13:36:23
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 14
Author PrimeFan (13766)
Entry type Definition
Classification msc 51-00
Synonym circular
Related topic SqueezingMathbbRn
Related topic CurvatureOfACircle
Defines unit circle
Defines radius
Defines radii
Defines perimeter
Defines circumference
Defines three point formula for the circle
Defines center