Circle

A circle 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 length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a disc.

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.

• 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 (or disk), while in everyday use the term "circle" may also refer to a disc.
• 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 \(r\) and required to be a positive number. A circle with \(r=0\) 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.
• 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.

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

Prehistoric people made stone circles and timber circles, and circular elements are common in petroglyphs and cave paintings. 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.

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

, Euclid, Book I, Elements

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. 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.

In 1880 CE, Ferdinand von Lindemann proved that π is transcendental, proving that the millennia-old problem of squaring the circle cannot be performed with straightedge and compass.

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.

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. Magic circles are part of some traditions of Western esotericism.

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

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, which comes to π multiplied by the radius squared: \[\mathrm{Area} = \pi r^2.\]

Equivalently, denoting diameter by d, \[\mathrm{Area} = \frac{\pi d^2}{4} \approx 0.7854 d^2,\] 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.

If a circle of radius r is centred at the vertex of an angle, and that angle intercepts an arc of the circle with an arc length of s, then the radian measure 𝜃 of the angle is the ratio of the arc length to the radius: \[\theta = \frac{s}{r}.\]

The circular arc is said to subtend the angle, known as the central angle, at the centre of the circle. One radian is the measure of the central angle subtended by a circular arc whose length is equal to its radius. The angle subtended by a complete circle at its centre is a complete angle, which measures 2π radians, 360 degrees, or one turn.

Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is \[s = \theta r,\]

and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is \[A = \frac{1}{2} \theta r^2.\]

In the special case 𝜃 = 2π, these formulae yield the circumference of a complete circle and area of a complete disc, respectively.

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 \[(x_1 - a)x + (y_1 - b)y = (x_1 - a)x_1 + (y_1 - b)y_1,\] or \[(x_1 - a)(x - a) + (y_1 - b)(y - b) = r^2.\]

If y₁ ≠ b, then the 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 \[x_1 x + y_1 y = r^2,\] and its slope is \[\frac{dy}{dx} = -\frac{x_1}{y_1}.\]

• 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 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.

• 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 a + b + c + d equals the square of the diameter.
• 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 8r − 4p, where r is the circle radius, and p is the distance from the centre point to the point of intersection.
• 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.

• 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.
• If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = ⁠1/2⁠arc(AQ).

• 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 AF = 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 (\(\overset{\frown}{DE}\) and \(\overset{\frown}{BC}\)). That is, \(2\angle{CAB} = \angle{DOE} - \angle{BOC}\), where O is the centre of the circle (secant, secant 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°).