In Euclidean geometry, a circle is the set of all points at a fixed distance, called the radius, from a fixed point, called the centre (center). The points can only be those that are part of a conic section; within the set of a plane bisecting a cone. Circles are simple closed curves, dividing the plane into an interior and exterior. Sometimes the word circle is used to mean the interior, with the circle itself called the circumference(C). Usually however, the circumference means the length of the circle, and the interior of the circle is called a disk or disc. diagram of a circle File links The following pages link to this file: Circle User:DrBob/Figures Image:Whale spread. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... A spatial point is an entity with a location in space but no extent (volume, area or length). ... The distance between two points is the length of a straight line segment between them. ... In mathematics, a conic section (or just conic) is a curved locus of points, fby intersecting a cone with a plane. ... This article is in need of attention from an expert on the subject. ... A cone is a basic geometrical shape: see cone (geometry). ... In mathematics, the concept of a curve tries to capture our intuitive idea of a geometrical one-dimensional and continuous object. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

Mathematical definitions

In an x-y coordinate system, the circle with centre (a, b) and radius r is the set of all points (x, y) such that See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ...

If the circle is centered at the origin (0, 0), then this formula can be simplified to In mathematics and in the sciences, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. ...

x² + y² = r².

The circle centered at the origin with radius 1 is called the unit circle. Illustration of a unit circle. ...

Expressed in parametric equations, (x, y) can be written as Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, parametric equations are a bit like functions: they allow someone to fill in some variables, called parameters or independent variables, with any values they wish. ...

x = a + r cos(t)

y = b + r sin(t).

The slope a circle at a point (x, y) can be expressed with the following formula, assuming the center is at the origin and (x, y) is on the circle: In mathematics, the slope (or gradient, especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the steepness of said line. ...

In the complex plane, a circle with a center at c and radius r has the equation | z − c | ² = r². Since , the slightly generalized equation for real p, q and complex g is sometimes called a generalized circle. It is important to note that not all generalized circles are actually circles. In mathematics, the complex plane is a way of visualising the space of the complex numbers. ...

All circles are similar; as a consequence, a circle's circumference and radius are proportional, as are its area and the square of its radius. The constants of proportionality are 2π and π, respectively. In other words: Several equivalence relations in mathematics are called similarity. ... The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. ... Area is a quantity expressing the size of a figure in Euclidean plane or surface. ... A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ... Lower-case pi The mathematical constant Ï€ is the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter and is commonly used in mathematics, physics, and engineering. ...

• Length of a circle's circumference =
• Area of a circle =

The formula for the area of a circle can be derived from the formula for the circumference and the formula for the area of a triangle, as follows. Imagine a regular hexagon (six-sided figure) divided into equal triangles, with their apices at the center of the hexagon. The area of the hexagon may be found by the formula for triangle area by adding up the lengths of all the triangle bases (on the exterior of the hexagon), multiplying by the height of the triangles (distance from the middle of the base to the center) and dividing by two. This is an approximation of the area of a circle. Then imagine the same exercise with an octagon (eight-sided figure), and the approximation is a little closer to the area of a circle. As a regular polygon with more and more sides is divided into triangles and the area calculated from this, the area becomes closer and closer to the area of a circle. In the limit, the sum of the bases approaches the circumference 2πr, and the triangles' height approaches the radius r. Multiplying the circumference and radius and dividing by 2, we get the area, π r². For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A regular hexagon A hexagon (also known as sexagon) is a polygon with six edges and six vertices. ... An approximation is an inexact representation of something that is still close enough to be useful. ... One of the 8 semi-regular tessellations: octagons and squares An octagon is a polygon that has eight sides. ... In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...

Properties

Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ...

Chord properties

• Chords equidistant from the centre of a circle are equal.
• Equal chords are equidistant from the centre.
• A line from the centre, perpendicular to a chord, bisects the chord.
• The line segment drawn from the centre to the midpoint of the chord is perpendicular to the chord.
• The perpendicular bisector of a chord passes through the centre of a circle.

Tangent properties

• The line drawn perpendicular to the end point of a radius is a tangent to the circle.
• A line drawn perpendicular to a tangent at the point of contact with a circle passes through the centre of the circle.
• Tangents drawn from a point outside the circle are equal in length.
• Two tangents can always be drawn from a point outside of the circle.

Inscribed angle theorem

• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
• If two angles are inscribed on the same chord and on the same side of the chord , then they are equal.
• An inscribed angle subtended by a semicircle is a right angle.
• For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

Secant, tangent, and chord properties

• The chord theorem states that if two chords, CD and EF, intersect at G, then . (Chord Theorem)
• If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then . (Tangent Secant Theorem)
• If two secants, DG and DE, also cut the circle at H and F respectively, then . (Corollary of the Tangent Secant Theorem)
• The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent Chord Property)
• If the angle subtended by the chord at the centre is 90 degrees then l = sqrt(2) * r, where l is the length of the chord and r is the radius of the circle.

The power of a point P with respect to a circle with center C and radius r is defined as Therefore points inside the circle have negative power, points outside have positive power, and points on the circle have power zero. ...

See also

• Unit circle
• Descartes' theorem
• Isoperimetric theorem
• List of circle topics

Illustration of a unit circle. ... In geometry, Descartes theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. ... Isoperimetry literally means having an equal perimeter. In mathematics, isoperimetry is the general study of geometric figures having equal boundaries. ... This list of circle topics is not intended for metaphorical circles, but rather for topics related to the geometric shape. ...

External links

• Clifford's Circle Chain Theorems. This is a step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression. by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
• Munching on Circles