*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*.

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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.

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.

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.

- 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 :

<math>r=\frac{y^2}{8x}+\frac{x}{2}</math>

- 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.

- The chord theorem states that if two chords, CD and EF, intersect at G, then <math>CG \times DG = EG \times FG</math>. (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 <math>DC^2 = DG \times DE</math>. (tangent-secant theorem) - If two secants, DG and DE, also cut the circle at H and F respectively, then <math>DH \times DG = DF \times DE</math>. (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.

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

- <math>

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

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

- <math>x^2 + y^2 = r^2 \!\ </math>

and its tangent will be

- <math>xx_1+yy_1=r^2 \!\ </math>

where <math>x_1</math>, <math>y_1</math> 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

- <math>x = a+r\,\cos t,\,\!</math>
- <math>y = b+r\,\sin t\,\!</math>

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

- <math>

ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0. </math>

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

- <math>

r^2 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2.\, </math>

In the complex plane, a circle with a center at *c* and radius *r* has the equation <math>|z-c|^2 = r^2</math>. Since <math>|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}</math>, the slightly generalized equation <math>pz\overline{z} + gz + \overline{gz} = q</math> 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.

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:

- <math>

y' = - \frac{x}{y}. </math>

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

- <math>

y' = \frac{a-x}{y-b}, </math>

provided that <math>y \neq b</math>, of course.

- The area enclosed by a circle is

- <math>

A = r^2 \cdot \pi = \frac{d^2\cdot\pi}{4} \approx 0{.}7854 \cdot d^2, </math>

that is, approximately 79 percent of the circumscribed square.

- Length of a circle's circumference is

- <math>

c = \pi d = 2\pi \cdot r. </math>

- Alternate formula for circumference:

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

- <math>

\frac{c}{A} = \frac{2 \pi r}{\pi r^2}. </math>

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

- <math>

\frac{c}{A} = \frac{2}{r}. </math>

Therefore solving for *c*:

- <math>

c = \frac{2A}{r} </math>

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.

The diameter of a circle is

- <math>

d = 2r= 2 \cdot \sqrt{\frac{A}{\pi}} \approx 1{.}1284 \cdot \sqrt{A}. </math>

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

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:

- <math>

\frac{AP}{BP} = \frac{AC}{BC} </math>

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to <math>180^{\circ}</math>, the angle CPD is exactly <math>90^{\circ}</math>, 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).

Given three non-collinear points lying on the circle

- <math>

\mathrm{P_1} = \begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix}, \mathrm{P_2} = \begin{bmatrix} x_2 \\ y_2 \\ z_2 \end{bmatrix}, \mathrm{P_3} = \begin{bmatrix} x_3 \\ y_3 \\ z_3 \end{bmatrix} </math>

The radius of the circle is given by

- <math>

\mathrm{r} = \frac {\left|P_1-P_2\right| \left|P_2-P_3\right|\left|P_3-P_1\right|} {2 \left|\left(P_1-P_2\right) \times \left(P_2-P_3\right)\right|} </math>

The center of the circle is given by

- <math>

\mathrm{P_c} = \alpha \, P_1 + \beta \, P_2 + \gamma \, P_3 </math>

where

- <math>

\alpha = \frac {\left|P_2-P_3\right|^2 \left(P_1-P_2\right) \cdot \left(P_1-P_3\right)} {2 \left|\left(P_1-P_2\right) \times \left(P_2-P_3\right)\right|^2} </math>

- <math>

\beta = \frac {\left|P_1-P_3\right|^2 \left(P_2-P_1\right) \cdot \left(P_2-P_3\right)} {2 \left|\left(P_1-P_2\right) \times \left(P_2-P_3\right)\right|^2} </math>

- <math>

\gamma = \frac {\left|P_1-P_2\right|^2 \left(P_3-P_1\right) \cdot \left(P_3-P_2\right)} {2 \left|\left(P_1-P_2\right) \times \left(P_2-P_3\right)\right|^2} </math>

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

- <math>

\hat{n} = \frac

{\left( P_2 - P_1 \right) \times \left(P_3-P_1\right)} {\left| \left( P_2 - P_1 \right) \times \left(P_3-P_1\right) \right|}

</math>

Given the radius, <math>\mathrm{r}</math> , center, <math>\mathrm{P_c}</math>, a point on the circle, <math>\mathrm{P_0}</math> and a unit normal of the plane containing the circle, <math>\hat{n}</math>, the parametric equation of the circle starting from the point <math>\mathrm{P_0}</math> and proceeding counterclockwise is given by the following equation:

- <math>

\mathrm{R} \left( s \right) = \mathrm{P_c} + \cos \left( \frac{\mathrm{s}}{\mathrm{r}} \right) \left( P_0 - P_c \right) + \sin \left( \frac{\mathrm{s}}{\mathrm{r}} \right) \left[ \hat{n} \times \left( P_0 - P_c \right) \right] </math>

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

- 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.

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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