Circles are simple shapes of Euclidean geometry consisting of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. A circle with center A is sometimes denoted by the symbol .

A chord of a circle is a line segment whose both endpoints lie on the circle. A diameter is a chord passing through the center. The length of a diameter is twice the radius. A diameter is the largest chord in a circle.

Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc is any connected part of a circle.

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

History

The circle has been known since before the beginning of recorded history. It is the basis for the wheel which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Some highlights in the history of the circle are:

• 1700BC - The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of $\backslash pi$.
• 300BC - Book 3 of Euclid's Elements deals with the properties of circles.
• 1880 Lindemann proves that $\backslash pi$ is transcendental, effectively settling the millennia old problem of squaring the circle.

Analytic results

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that

\left( x - a \right)^2 + \left( y - b \right)^2=r^2.

The equation of the circle follows from the Pythagorean theorem applied to any point on the circle. If the circle is centred at the origin (0, 0), then this formula can be simplified to

$x^2\; +\; y^2\; =\; r^2.\; \backslash !\backslash$

When expressed in parametric equations, (x, y) can be written using the trigonometric functions sine and cosine as

$x\; =\; a+r\backslash ,\backslash cos\; t,\backslash ,\backslash !$

$y\; =\; b+r\backslash ,\backslash sin\; t\backslash ,\backslash !$

where t is a parametric variable, understood by many as the angle the ray to (x, y) makes with the x-axis. Alternatively, in stereographic coordinates, the circle has a parametrization