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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':
<br /> x^2 + y^2 = r^2<br />
Equation for a circle with center at (a, b) and radius 'r':
<br /> (x - a)^2 + (y - b)^2 = r^2<br />
General equation (expanded) for a circle:
<br /> x^2 + y^2 + 2fx + 2gy + c = 0<br />
The center of such a circle is given as (-f, -g) and the radius 'r' is given by:
<br /> r = \sqrt{g^2 + f^2 - c}<br />
The slope of the tangent at a point 'x' for a circle centered at (a, b) is given as:
<br /> \tan \theta = -\frac{x - a}{y - b}<br />
Area of a circle is given as:
<br /> A = \pi r^2<br />
Length of an arc subtending an angle \theta is given as:
<br /> L = r\theta<br />
For the length of the complete circle, \theta = 2\pi
For a circle, centered at origin, and radius 'r' a point on the circle, the radius to which makes an angle \theta with the positive x-axis is given as:
<br /> x = r\sin \theta<br />
<br /> y = r\cos \theta<br />
Extended explanation
Other definitions from more general concepts:
1. a special instance of an ellipse, having an eccentricity, e = 0 (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!
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':
<br /> x^2 + y^2 = r^2<br />
Equation for a circle with center at (a, b) and radius 'r':
<br /> (x - a)^2 + (y - b)^2 = r^2<br />
General equation (expanded) for a circle:
<br /> x^2 + y^2 + 2fx + 2gy + c = 0<br />
The center of such a circle is given as (-f, -g) and the radius 'r' is given by:
<br /> r = \sqrt{g^2 + f^2 - c}<br />
The slope of the tangent at a point 'x' for a circle centered at (a, b) is given as:
<br /> \tan \theta = -\frac{x - a}{y - b}<br />
Area of a circle is given as:
<br /> A = \pi r^2<br />
Length of an arc subtending an angle \theta is given as:
<br /> L = r\theta<br />
For the length of the complete circle, \theta = 2\pi
For a circle, centered at origin, and radius 'r' a point on the circle, the radius to which makes an angle \theta with the positive x-axis is given as:
<br /> x = r\sin \theta<br />
<br /> y = r\cos \theta<br />
Extended explanation
Other definitions from more general concepts:
1. a special instance of an ellipse, having an eccentricity, e = 0 (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!