What is the equation of a circle

[Greg Bernhardt]

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':
$$x^2 + y^2 = r^2$$

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

General equation (expanded) for a circle:
$$x^2 + y^2 + 2fx + 2gy + c = 0$$

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

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

Area of a circle is given as:
$$A = \pi r^2$$

Length of an arc subtending an angle $\theta$ is given as:
$$L = r\theta$$

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:
$$x = r\sin \theta$$
$$y = r\cos \theta$$

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.

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[fresh_42]

See also: https://www.physicsforums.com/threads/what-is-a-conic.762859/