Circle: Encyclopedia - Circle

Circle

A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. 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. An arc is any continuous portion of a circle.

Circle - 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

If the circle is centred at the origin (0, 0), then this formula can be simplified to

The circle centred at the origin with radius 1 is called the unit circle.

Expressed in parametric equations, (x, y) can be written as

The slope a circle at a point (x, y) can be expressed with the following formula, assuming the centre is at the origin and (x, y) is on the circle:

In the complex plane, a circle with a centre 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.

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:

• 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 centre 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.

Circle - Properties

Circle - 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.

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.

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.

Circle - 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.

See also

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

Adapted from the Wikipedia article "Circle", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki