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 called the centre (or center; cf. American and British English spelling differences). The common distance of the points of a circle from its centre is called its radius.

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 (also known as the perimeter) or to the whole figure including its interior. However, in strict technical usage, "circle" refers to the perimeter while the interior of the circle is called a disk. The circumference of a circle is the perimeter of the circle (especially when referring to its length).

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

Further terminology

Chord, secant, tangent, and diameter.

Arc, sector, and segment

The diameter of a circle is the length of a line segment whose endpoints lie on the circle and which passes through the centre of the circle. This is the largest distance between any two points on the circle. The diameter of a circle is twice its radius.

As well as referring to lengths, the terms "radius" and "diameter" can also refer to actual line segments (respectively, a line segment from the centre of a circle to its perimeter, and a line segment between two points on the perimeter passing through the centre). In this sense, the midpoint of a diameter is the centre and so it is composed of two radii.

A chord of a circle is a line segment whose two endpoints lie on the circle. The diameter, passing through the circle's centre, is the longest chord in a circle. A tangent to a circle is a straight line that touches the circle at a single point. 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.

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 circle has been known since before the beginning of recorded history. It 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 and calculus.

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.

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 $\pi$ .^[1]
300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
1880 – Lindemann proves that $\pi$ is transcendental, effectively settling the millennia-old problem of squaring the circle.^[2]

Analytic results

Length of circumference

For more details on this topic, see Pi.

The ratio of a circle's circumference to its diameter is π (pi), a constant that takes the same value (approximately 3.141592654) for all circles. Thus the length of the circumference (c) is related to the radius (r) by

$c = 2 \pi r\,$

or equivalently to the diameter (d) by

$c = \pi d.\,$

Using the circle constant $\tau=2\pi$ one gets

$c = \tau r .\,$

Area enclosed

Area of the circle = π × area of the shaded square

As proved by Archimedes the area of the area enclosed by a circle is

$Area = \frac{1}{2}r\cdot c$ .

Equivalently, the area is $\pi$ multiplied by the radius squared:

$Area = \pi r^2.\,$

Using the circle constant $\tau=2\pi$ one gets

$Area = \frac{1}{2}\tau r^2.\,$

Equivalently, denoting diameter by d,

$Area = \frac{\pi d^2}{4} \approx 0{.}7854 \cdot d^2,$

that is, approximately 79% of the circumscribing square (whose side is of length d).

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 (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 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 (x, y) 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 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

$r = 2 a\cos(\theta - \phi)$ .

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 $z = re^{it}+c$ .

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 (x₁−a)x+(y₁−b)y = c. Evaluating at (x₁, y₁) determines the value of c and the result is that the equation of the tangent is

$(x_1-a)x+(y_1-b)y = (x_1-a)x_1+(y_1-b)y_1$

$(x_1-a)(x-a)+(y_1-b)(y-b) = r^2$ .

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

$x_1x+y_1y = r^2$ ,

and its slope is

$\frac{dy}{dx} = -\frac{x_1}{y_1}$ .

Properties

The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
All circles are similar.
- A circle's circumference and radius are proportional.
- The area enclosed and the square of its radius are proportional.
  - The constants of proportionality are 2π and π, respectively.
The circle which is centred at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

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.

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 drawn perpendicular 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.

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^[3].

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.

Notes

↑ Chronology for 30000 BC to 500 BC
↑ Squaring the circle
↑ Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co.. pp. 30. http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002.

References

Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.
"Circle" in The MacTutor History of Mathematics archive

External links

Circle (PlanetMath.org website)
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
Munching on Circles at cut-the-knot
Ron Blond homepage - interactive applets
calculate circumference and area with your own values
MathAce's Circle article - has a good in-depth explanation of unit circles and transforming circular equations.
Google Maps Circle Overlay Lets you add a circle to Google Maps