# Circle

*This article is about the shape and mathematical concept of circle. For other uses of the term, see Circle (disambiguation).*

In Euclidean geometry, a **circle** is the set of all points in a plane at a fixed distance, called the *radius*, from a given point, the *center*. The length of the circle is called its *circumference*, and any continuous portion of the circle is called an *arc*.

A circle is a simple closed curve that divides the plane into an interior and exterior. The interior of the circle is called a *disk*.

## Contents |

Mathematically, a circle can be understood in several other ways as well. For instance, it is a special case of an ellipse in which the two foci coincide (that is, they are the same point). Alternatively, a circle can be thought of as the conic section attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

## Properties

All circles have similar properties. Some of these are noted below.

- For any circle, the area enclosed and the square of its radius are in a fixed proportion, equal to the mathematical constant π.
- For any circle, the circumference and radius are in a fixed proportion, equal to 2π.
- The circle is the shape with the highest area for a given length of perimeter.
- The circle is a highly symmetrical shape. Every line through the center forms a line of reflection symmetry. In addition, there is rotational symmetry around the center for every angle. The symmetry group is called the orthogonal group O(2,
**R**), and the group of rotations alone is called the circle group**T**. - The circle centered at the origin with radius 1 is called the unit circle.

### Chord properties

A line segment that connects one point of a circle to another is called a *chord*. The *diameter* is a chord that runs through the center of the circle.

- The diameter is longest chord of the circle.
- Chords equidistant from the center of a circle are equal in length. Conversely, chords that are equal in length are equidistant from the center.
- A line drawn through the center of a circle perpendicular to a chord bisects the chord. Alternatively, one can state that a line drawn through the center of a circle bisecting a chord is perpendicular to the chord. This line is called the
*perpendicular bisector*of the chord. Thus, one could also state that the perpendicular bisector of a chord passes through the center of the circle. - 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.
- If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
- An inscribed angle subtended by a diameter is a right angle.

### Sagitta properties

- The sagitta is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference of the circle.
- Given the length of a chord,
**y**, and the length**x**of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the 2 lines :

### 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 center 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.

### Theorems

- 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 center is 90 degrees then
*l*= √(2) ×*r*, where*l*is the length of the chord and*r*is the radius of the circle. - If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

## Analytic results

### Equation of a circle

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

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

and its tangent will be

where *x*_{1}, *y*_{1} are the coordinates of the common point.

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

where *t* is a parametric variable, understood as the angle the ray to (*x*, *y*) makes with the *x*-axis.

In homogeneous coordinates each conic section with equation of a circle is

*a**x*^{2}+*a**y*^{2}+ 2*b*_{1}*x**z*+ 2*b*_{2}*y**z*+*c**z*^{2}= 0.

It can be proven that a *conic section* is a circle if and only if the point I(1,i,0) and J(1,-i,0) lie on the conic section. These points are called the circular points at infinity.

In polar coordinates the equation of a circle is

In the complex plane, a circle with a center at *c* and radius *r* has the equation | *z* − *c* | ^{2} = *r*^{2}. 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.

### Slope

The slope of 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:

More generally, the slope at a point (*x*, *y*) on the circle (*x* − *a*)^{2} + (*y* − *b*)^{2} = *r*^{2}, (i.e., the circle centered at [*a*, *b*] with radius *r* units), is given by

provided that , of course.

### Area enclosed

- The area enclosed by a circle is

that is, approximately 79 percent of the circumscribed square.

### Circumference

- Length of a circle's circumference is

- Alternate formula for circumference:

Given that the ratio circumference *c* to the Area *A* is

The *r* and the π can be canceled, leaving

Therefore solving for *c*:

So the circumference is equal to 2 times the area, divided by the radius. This can be used to calculate the circumference when a value for π cannot be computed.

### Diameter

The diameter of a circle is

## Inscribed angles

An inscribed angle ψ is exactly half of the corresponding central angle θ (see Figure). Hence, all inscribed angles that subtend the same arc have the same value (cf. the blue and green angles ψ in the Figure). Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle.

## An alternative definition of a circle

Apollonius of Perga showed that a circle may also be defined as the set of points having a constant *ratio* of distances to two foci, A and B.

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to , the angle CPD is exactly , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

As a point of clarification, note that C and D are determined by A, B, and the desired ratio (i.e. A and B are not arbitrary points lying on an extension of the diameter of an existing circle).

## Calculating the parameters of a circle

Given three non-collinear points lying on the circle

### Radius

The radius of the circle is given by

### Center

The center of the circle is given by

where

### Plane unit normal

A unit normal of the plane containing the circle is given by

### Parametric Equation

Given the radius, r , center, P_{c}, a point on the circle, P_{0} and a unit normal of the plane containing the circle, , the parametric equation of the circle starting from the point P_{0} and proceeding counterclockwise is given by the following equation:

## See also

- Area of a disk
- Ball (mathematics)
- Degree (angle)
- Ellipse
- Pi
- Sphere

## References

- Altshiller-Court, Nathan. 2007.
*College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle*. New York, NY: Dover Publications. ISBN 0486458059.

- Arnone, Wendy. 2001.
*Geometry for Dummies*. Hoboken, NJ: For Dummies (Wiley). ISBN 0764553240.

- Pedoe, Dan. 1997.
*Circles: A Mathematical View*. Washington, DC: The Mathematical Association of America. ISBN 0883855186.

## External links

All links retrieved May 22, 2013.

- Interactive Java applets – for the properties of and elementary constructions involving circles.
- Interactive Standard Form Equation of Circle – Click and drag points to see standard form equation in action.
- Clifford's Circle Chain Theorems. – 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".
- What Is Circle? – at cut-the-knot.
- Ron Blond homepage - interactive applets.

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