(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# What is the equation of a circle

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