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# Circle

Circle illustration showing a radius, a diameter, the centre and the circumference
Tycho crater, one of many examples of circles that arise in nature. NASA photo
A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are equidistant from a given point, the centre.
Circles are simple closed curves which divide the plane into two regions, an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk.
A circle is a special ellipse in which the two foci are coincident and the eccentricity is 0. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

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## Terminology

A circle's diameter is the length of a line segment whose endpoints lie on the circle and which passes through the centre. This is the largest distance between any two points on the circle. The diameter of a circle is twice the radius, or distance from the centre to the circle's boundary (circumference). The terms 'diameter' and 'radius' also refer to the line segments which fit these descriptions.
A chord is a line segment whose endpoints lie on the circle. A diameter is the longest chord in a circle. A tangent to a circle is a straight line that touches the circle at a single point, while a secant is an extended chord: a straight line cutting the circle at two points.
An arc of a circle is any connected part of the circle's circumference. A sector is a region bounded by two radii and an arc lying between the radii, and a segment is a region bounded by a chord and an arc lying between the chord's endpoints.
 Chord, secant, tangent, and diameter. Arc, sector, and segment

## History

The compass in this 13th century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo
The etymology of the word circle is from the Greek, kirkos "a circle," from the base ker- which means to turn or bend. The origin of the word "circus" is closely related as well.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle 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, astronomy, and calculus.
Circles on an old astronomy drawing
Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[citation needed]
Some highlights in the history of the circle are:
• 1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 (3.16049...) as an approximate value of π.[1]
• 300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
• In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.
• 1880 – Lindemann proves that π is transcendental, effectively settling the millennia-old problem of squaring the circle.[2]

## Analytic results

### Length of circumference

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant (approximately equal to 3.141592654). Thus the length of the circumference C is related to the radius r and diameter d by C = 2πr = πd.

### Area enclosed

Area of the circle = π × area of the shaded square
As proved by Archimedes, the area enclosed by a circle is π multiplied by the radius squared:
$\mathrm{Area} = \pi r^2.\,$
Equivalently, denoting diameter by d,
$\mathrm{Area} = \frac{\pi d^2}{4} \approx 0{.}7854d^2,$
that is, approximately 79% of the circumscribing square (whose side is of length d). These results can be obtained in several ways, including via integration and by considering the circle as a regular polygon with an infinite number of sides.
The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

### Equations

#### Cartesian coordinates

Circle of radius r = 1, centre (a, b) = (1.2, -0.5)
In an x-y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that
$\left(x - a \right)^2 + \left( y - b \right)^2=r^2.$
This equation of the circle follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length x − a and y − b. If the circle is centred at the origin (0, 0), then the equation simplifies to
$x^2 + y^2 = r^2. \!\$
The equation can be written in parametric form using the trigonometric functions sine and cosine in a polar coordinate system as
$x = a+r\,\cos t,\,\!$
$y = b+r\,\sin t\,\!$
where t is a parametric variable, interpreted geometrically as the angle that the ray from the origin to (xy) makes with the x-axis. Alternatively, a rational parametrization of the circle is:
$x = a + r \frac{1-t^2}{1+t^2}$
$y = b + r \frac{2t}{1+t^2}.$
In this parametrization, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.
In homogeneous coordinates each conic section with equation of a circle is of the form
$\ ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.$
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.

#### Polar coordinates

In polar coordinates the equation of a circle is:
$r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2\,$
where a is the radius of the circle, r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle. For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes
r = 2acos(θ − φ).
In the general case, the equation can be solved for r, giving
$r = r_0 \cos(\theta - \phi) + \sqrt{a^2 - r_0^2 \sin^2(\theta - \phi)}$,
the solution with a minus sign in front of the square root giving the same curve.

#### Complex plane

In the complex plane, a circle with a centre at c and radius (r) has the equation $|z-c|^2 = r^2\,$. In parametric form this can be written z = reit + c.
The slightly generalised equation $pz\overline{z} + gz + \overline{gz} = q$ for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with $p = 1,\ g=\overline{c},\ q=r^2-|c|^2$, since $|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}$. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

### Tangent lines

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1a)x+(y1b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is
(x1a)x + (y1b)y = (x1a)x1 + (y1b)y1
or
(x1a)(xa) + (y1b)(yb) = r2.
If y1≠b then slope of this line is
$\frac{dy}{dx} = -\frac{x_1-a}{y_1-b}$.
This can also be found using implicit differentiation.
When the centre of the circle is at the origin then the equation of the tangent line becomes
x1x + y1y = r2,
and its slope is
$\frac{dy}{dx} = -\frac{x_1}{y_1}$.

## Properties

### Chord

• Chords are equidistant from the centre of a circle if and only if they are equal in length.
• The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
• A perpendicular line from the centre of a circle bisects the chord.
• The line segment (circular segment) through the centre bisecting a chord is perpendicular to the chord.
• 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.
• For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
• An inscribed angle subtended by a diameter is a right angle (see Thales' theorem).
• The diameter is the longest chord of the circle.
• If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd.
• If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.
• The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.[3]:p.71

### Sagitta

• The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle.
• Given the length y of a chord, 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 two lines:
$r=\frac{y^2}{8x}+ \frac{x}{2}.$
Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y/2)². Solving for r, we find the required result.

### Tangent

• The line perpendicular drawn to a radius through the end point of the radius is a tangent to the circle.
• A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle.
• Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.
• If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the center as O, the angles ∠BOA and ∠BPA are supplementary.
• If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = $\tfrac{1}{2}$arc(AQ).

### Theorems

Secant-secant theorem
• The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA.
• 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 DC2 = DG×DE. (Tangent-secant theorem.)
• If two secants, DG and DE, also cut the circle at H and F respectively, then DH×DG = DF×DE. (Corollary of the tangent-secant theorem.)
• The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of the chord (Tangent Chord Angle).
• If the angle subtended by the chord at the centre 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.

### Inscribed angles

Inscribed angle theorem
An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

## Apollonius circle

Apollonius' definition of a circle: d1/d2 constant
Apollonius of Perga showed that a circle may also be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B. (The set of points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn about two points[4].
The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:
$\frac{AP}{BP} = \frac{AC}{BC}.$
Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{\circ}$, the angle CPD is exactly $90^{\circ}$, 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.

### Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the Apollonius circle for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:
$|[A,B;C,P]| = 1.\$
Stated another way, P is a point on the Apollonius circle if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane.

### Generalized circles

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition
$\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}$   (1)
is not a circle, but rather a line.
Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying (1) is called a generalized circle. It may either be a true circle or a line. In this sense a line is a generalized circle of infinite radius.