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What is the equation of a circle

  1. Jul 23, 2014 #1
    Definition/Summary

    A circle has many definitions, a classical one being "the locus of all points on a plane that are equidistant from a given point, which is referred to as the 'center' of the circle".

    Equations

    Equation for a circle with it's center as the origin and radius 'r':
    [tex]
    x^2 + y^2 = r^2
    [/tex]

    Equation for a circle with center at (a, b) and radius 'r':
    [tex]
    (x - a)^2 + (y - b)^2 = r^2
    [/tex]

    General equation (expanded) for a circle:
    [tex]
    x^2 + y^2 + 2fx + 2gy + c = 0
    [/tex]

    The center of such a circle is given as (-f, -g) and the radius 'r' is given by:
    [tex]
    r = \sqrt{g^2 + f^2 - c}
    [/tex]

    The slope of the tangent at a point 'x' for a circle centered at (a, b) is given as:
    [tex]
    \tan \theta = -\frac{x - a}{y - b}
    [/tex]

    Area of a circle is given as:
    [tex]
    A = \pi r^2
    [/tex]

    Length of an arc subtending an angle [itex]\theta[/itex] is given as:
    [tex]
    L = r\theta
    [/tex]

    For the length of the complete circle, [itex]\theta = 2\pi[/itex]

    For a circle, centered at origin, and radius 'r' a point on the circle, the radius to which makes an angle [itex]\theta[/itex] with the positive x-axis is given as:
    [tex]
    x = r\sin \theta
    [/tex]
    [tex]
    y = r\cos \theta
    [/tex]

    Extended explanation

    Other definitions from more general concepts:

    1. a special instance of an ellipse, having an eccentricity, [itex]e = 0[/itex] (i.e., equal lengths of major and minor axes),

    2. a conic section formed by the intersection of a cone with a plane normal to its axis of symmetry,

    3. the two-dimensional instance of an n-dimensional hypersphere.

    * This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!
     
  2. jcsd
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