Contents

Circle

Circle
A circle   circumference C   diameter D   radius R   centre or origin O
Type	Conic section
Symmetry group	O(2)
Area	πR2
Perimeter	C = 2πR

Geometry
Projectingasphere to a plane
Outline History (Timeline)
Branches Euclidean Non-Euclidean Elliptic Spherical Hyperbolic Non-Archimedean geometry Projective Affine Synthetic Analytic Algebraic Arithmetic Diophantine Differential Riemannian Symplectic Discrete differential Complex Finite Discrete/Combinatorial Digital Convex Computational Fractal Incidence Noncommutative geometry Noncommutative algebraic geometry
Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
One-dimensional Line segment ray Length
Two-dimensional Plane Area Polygon Triangle Altitude Hypotenuse Pythagorean theorem Parallelogram Square Rectangle Rhombus Rhomboid Quadrilateral Trapezoid Kite Circle Diameter Circumference Area
Three-dimensional Volume Cube cuboid Cylinder Dodecahedron Icosahedron Octahedron Pyramid Platonic Solid Sphere Tetrahedron
Four- / other-dimensional Tesseract Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski Minggatu Pascal Pythagoras Parameshvara Poincaré Riemann Sakabe Sijzi al-Tusi Veblen Virasena Yang Hui al-Yasamin Zhang List of geometers
by period BCE Ahmes Baudhayana Manava Pythagoras Euclid Archimedes Apollonius 1–1400s Zhang Kātyāyana Aryabhata Brahmagupta Virasena Alhazen Sijzi Khayyám al-Yasamin al-Tusi Yang Hui Parameshvara 1400s–1700s Jyeṣṭhadeva Descartes Pascal Huygens Minggatu Euler Sakabe Aida 1700s–1900s Gauss Lobachevsky Bolyai Riemann Klein Poincaré Hilbert Minkowski Cartan Veblen Coxeter Present day Atiyah Gromov
v t e

Acircle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. The distance between any point of the circle and the centre is called the radius.

The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

Terminology

• Annulus: a ring-shaped object, the region bounded by two concentric circles.
• Arc: any connected part of a circle. Specifying two end points of an arc and a centre allows for two arcs that together make up a full circle.
• Centre: the point equidistant from all points on the circle.
• Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into two segments.
• Circumference: the length of one circuit along the circle, or the distance around the circle.
• Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
• Disc: the region of the plane bounded by a circle. In strict mathematical usage, a circle is only the boundary of the disc, while in everyday the terms "circle" and "disc" may be used interchangeably.
• Lens: the region common to (the intersection of) two overlapping discs.
• Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is half (the length of) a diameter. Usually, the radius is denoted ${\displaystyle r}$ and required to be a positive number. A circle with ${\displaystyle r=0}$ is a degenerate case consisting of a single point.
• Sector: a region bounded by two radii of equal length with a common centre and either of the two possible arcs, determined by this centre and the endpoints of the radii.
• Segment: a region bounded by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions not containing the centre of the circle to which their arc belongs to.
• Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.
• Semicircle: one of the two possible arcs determined by the endpoints of a diameter, taking its midpoint as centre. In non-technical common usage it may mean the interior of the two-dimensional region bounded by a diameter and one of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
• Tangent: a coplanar straight line that has one single point in common with a circle ("touches the circle at this point").

All of the specified regions may be considered as open, that is, not containing their boundaries, or as closed, including their respective boundaries.

Etymology

The word circle derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring".^[1] The origins of the words circus and circuit are closely related.

History

Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings.^[2] Disc-shaped prehistoric artifacts include the Nebra sky disc and jade discs called Bi.

The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle. The result corresponds to 256/81 (3.16049...) as an approximate value of π.^[3]

Book 3 of Euclid's Elements deals with the properties of circles. Euclid's definition of a circle is:

A circle is a plane figure bounded by one curved line, and such that all straight lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre.

— Euclid, Elements, Book I^[4]: 4

InPlato'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. 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.^[5][6]

In 1880 CE, Ferdinand von Lindemann proved that πistranscendental, proving that the millennia-old problem of squaring the circle cannot be performed with straightedge and compass.^[7]

With the advent of abstract art in the early 20th century, geometric objects became an artistic subject in their own right. Wassily Kandinsky in particular often used circles as an element of his compositions.^[8][9]

Symbolism and religious use

From the time of the earliest known civilisations – such as the Assyrians and ancient Egyptians, those in the Indus Valley and along the Yellow River in China, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist's message and to express certain ideas. However, differences in worldview (beliefs and culture) had a great impact on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their democratic manifestation, others focused on its centre to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, but in religious traditions it represents heavenly bodies and divine spirits.

The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, eye, aureole, mandorla, etc.), the ouroboros, the Dharma wheel, a rainbow, mandalas, rose windows and so forth.^[10] Magic circles are part of some traditions of Western esotericism.

Analytic results

Circumference

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference C is related to the radius r and diameter d by: $${\displaystyle C=2\pi r=\pi d.}$$

Area enclosed

As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,^[11] which comes to π multiplied by the radius squared: $${\displaystyle \mathrm {Area} =\pi r^{2}.}$$

Equivalently, denoting diameter by d, $${\displaystyle \mathrm {Area} ={\frac {\pi d^{2}}{4}}\approx 0.7854d^{2},}$$ that is, approximately 79% of the circumscribing square (whose side is of length d).

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

Equation of a circle

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 $${\displaystyle (x-a)^{2}+(y-b)^{2}=r^{2}.}$$

This equation, known as the equation of the circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the adjacent diagram, 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 $${\displaystyle x^{2}+y^{2}=r^{2}.}$$

Parametric form

The equation can be written in parametric form using the trigonometric functions sine and cosine as $${\displaystyle {\begin{aligned}x&=a+r\,\cos t,\\y&=b+r\,\sin t,\end{aligned}}}$$ where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x axis.

An alternative parametrisation of the circle is $${\displaystyle {\begin{aligned}x&=a+r{\frac {1-t^{2}}{1+t^{2}}},\\y&=b+r{\frac {2t}{1+t^{2}}}.\end{aligned}}}$$

In this parameterisation, the ratio of ttor can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x axis (see Tangent half-angle substitution). However, this parameterisation works only if t is made to range not only through all reals but also to a point at infinity; otherwise, the leftmost point of the circle would be omitted.

3-point form

The equation of the circle determined by three points ${\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),(x_{3},y_{3})}$ not on a line is obtained by a conversion of the 3-point form of a circle equation: $${\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {red}y}-y_{1})({\color {red}y}-y_{2})}{({\color {red}y}-y_{1})({\color {green}x}-x_{2})-({\color {red}y}-y_{2})({\color {green}x}-x_{1})}}={\frac {(x_{3}-x_{1})(x_{3}-x_{2})+(y_{3}-y_{1})(y_{3}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{3}-x_{1})}}.}$$

Homogeneous form

Inhomogeneous coordinates, each conic section with the equation of a circle has the form $${\displaystyle x^{2}+y^{2}-2axz-2byz+cz^{2}=0.}$$

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.

Polar coordinates

Inpolar coordinates, the equation of a circle is $${\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{2}=a^{2},}$$

where a is the radius of the circle, ${\displaystyle (r,\theta )}$ are the polar coordinates of a generic point on the circle, and ${\displaystyle (r_{0},\phi )}$ are the polar coordinates of the centre of the circle (i.e., 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 on the origin, i.e. r₀ = 0, this reduces to r = a. When r₀ = a, or when the origin lies on the circle, the equation becomes $${\displaystyle r=2a\cos(\theta -\phi ).}$$

In the general case, the equation can be solved for r, giving $${\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{2}\sin ^{2}(\theta -\phi )}}.}$$ Without the ± sign, the equation would in some cases describe only half a circle.

Complex plane

In the complex plane, a circle with a centre at c and radius r has the equation $${\displaystyle |z-c|=r.}$$

In parametric form, this can be written as $${\displaystyle z=re^{it}+c.}$$

The slightly generalised equation $${\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}$$

for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with ${\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}$, since ${\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}$. 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 $${\displaystyle (x_{1}-a)x+(y_{1}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{1},}$$or$${\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}$$

Ify₁ ≠ b, then the slope of this line is $${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{1}-b}}.}$$

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 $${\displaystyle x_{1}x+y_{1}y=r^{2},}$$ and its slope is $${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}$$

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.^[12]
  • A circle 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 that 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 are:
  • A perpendicular line from the centre of a circle bisects the chord.
  • The line 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 supplementary.
  • 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.
  • Among all the circles with a chord AB in common, the circle with minimal radius is the one with diameter AB.
• 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.^[13]
• The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point and is given by 8r² − 4p², where r is the circle radius, and p is the distance from the centre point to the point of intersection.^[14]
• 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.^[15]: p.71

Tangent

• A line drawn perpendicular to a radius through the end point of the radius lying on the circle 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 centre as O, the angles ∠BOA and ∠BPA are supplementary.
• IfAD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = 1/2arc(AQ).

Theorems

• The chord theorem states that if two chords, CD and EB, intersect at A, then AC × AD = AB × AE.
• If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem).
• A tangent can be considered a limiting case of a secant whose ends are coincident. If a tangent from an external point A meets the circle at F and a secant from the external point A meets the circle at C and D respectively, then AF² = AC × AD (tangent–secant theorem).
• The angle between a chord and the tangent at one of its endpoints is equal to one half the angle subtended at the centre of the circle, on the opposite side of the chord (tangent chord angle).
• If the angle subtended by the chord at the centre is 90°, then ℓ = r √2, 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 (${\displaystyle {\overset {\frown }{DE}}}$ and ${\displaystyle {\overset {\frown }{BC}}}$). That is, ${\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}$, where O is the centre of the circle (secant–secant theorem).

Inscribed angles

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°).

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 that will fit around the two lines: $${\displaystyle r={\frac {y^{2}}{8x}}+{\frac {x}{2}}.}$$

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 that it is a 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.

Compass and straightedge constructions

There are many compass-and-straightedge constructions resulting in circles.

The simplest and most basic is the construction given the centre of the circle and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the point on the circle and rotate the compass.

Construction with given diameter

• Construct the midpoint M of the diameter.
• Construct the circle with centre M passing through one of the endpoints of the diameter (it will also pass through the other endpoint).

Construction through three noncollinear points

• Name the points P, Q and R,
• Construct the perpendicular bisector of the segment PQ.
• Construct the perpendicular bisector of the segment PR.
• Label the point of intersection of these two perpendicular bisectors M. (They meet because the points are not collinear).
• Construct the circle with centre M passing through one of the points P, QorR (it will also pass through the other two points).

Circle of Apollonius

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.^[16][17] (The set of points where the distances are equal is the perpendicular bisector of segment AB, a line.) That circle is sometimes said to be drawn about two points.

The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point P satisfying the ratio of distances must fall on a particular circle. Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB, since the segments are similar: $${\displaystyle {\frac {AP}{BP}}={\frac {AC}{BC}}.}$$