Circle

A circle is a shape consisting of all points in a plane that are at a given distance, the radius, from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. 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, and is the basis for a vast range of geometry, trigonometry, astronomy, and engineering. A circle is a simple closed curve which divides 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 only the boundary and the whole figure is called a disc. The circle is a fundamental object in Euclidean geometry, and its properties give rise to an immense body of theorems and practical applications.

History and Philosophy

The circle has been regarded as a symbol of perfection, unity, and eternity across many cultures. Natural circles, such as the Sun and the Moon, were observed by early humans and likely inspired the first deliberate constructions of circular shapes. Archaeological findings, including stone circles and pottery, demonstrate that the circle was a known concept in prehistoric times. In ancient Greece, the circle was studied intensely by geometers. Thales of Miletus and the Pythagoreans investigated its properties, and Euclid devoted Book III of his Elements to the geometry of the circle, establishing a rigorous axiomatic foundation. Plato considered the circle the most perfect shape, and later, Aristotelian cosmology placed the Earth at the centre of a universe composed of concentric celestial spheres moving in perfect circles. This model dominated astronomy until the work of Johannes Kepler, who demonstrated that planetary orbits are elliptical. The problem of squaring the circle—constructing a square with the same area as a given circle using only a compass and straightedge—preoccupied mathematicians for millennia until it was proven impossible in 1882, following the demonstration that pi is a transcendental number. The circle remains a central object in mathematics, from complex analysis to differential geometry.

Definitions and Terminology

A circle can be formally defined in several equivalent ways. It is the locus of points at a fixed distance from a centre. A circle of radius \( r \) centred at the origin is the set of all points \((x, y)\) satisfying \( x^2 + y^2 = r^2 \). Alternatively, a circle is a special case of an ellipse in which the two foci coincide, and it is the conic section obtained by the intersection of a right circular cone with a plane perpendicular to the cone’s axis.

Key terminology associated with the circle includes the following:

• Centre: The fixed point equidistant from all points on the circle.
• Radius: A line segment from the centre to any point on the circle, or the length of such a segment. The plural is radii.
• Diameter: A line segment passing through the centre whose endpoints lie on the circle. Its length is twice the radius. It is the longest chord of a circle.
• Chord: A line segment whose endpoints both lie on the circle.
• Secant: A line that intersects the circle at two points.
• Tangent: A line that touches the circle at exactly one point, called the point of tangency. The radius to the point of tangency is perpendicular to the tangent.
• Arc: Any connected part of the circumference. A semicircle is an arc whose endpoints are the endpoints of a diameter.
• Sector: The region bounded by two radii and the intercepted arc.
• Segment: The region bounded by a chord and the arc subtended by that chord.
• Circumference: The length of the circle’s boundary.

Analytic Geometry

In a Cartesian coordinate system, the circle with centre at \((h, k)\) and radius \( r \) is the set of all points \((x, y)\) such that:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

This equation follows directly from the distance formula. When the centre is at the origin, the equation simplifies to \( x^2 + y^2 = r^2 \). The circle can also be expressed in parametric form. For a circle centred at the origin, the parametric equations are:

\[
x = r \cos t, \quad y = r \sin t
\]

where \( t \) is the parameter, typically representing the angle measured from the positive x‑axis, ranging from 0 to \( 2\pi \). For a general centre \((h, k)\), the equations become \( x = h + r\cos t \), \( y = k + r\sin t \).

In polar coordinates, the equation of a circle centred at the pole (origin) is simply \( \rho = r \). More generally, for a circle with centre at \((\rho_0, \theta_0)\) and radius \( R \), the polar equation is \( \rho^2 - 2\rho\rho_0\cos(\theta - \theta_0) + \rho_0^2 = R^2 \). In the complex plane, a circle with centre \( c \) and radius \( r \) can be described by the equation \( |z - c| = r \), where \( z \) is a complex variable.

Properties

Symmetry is a fundamental property of the circle. A circle has infinitely many lines of symmetry: any line passing through the centre. It also possesses rotational symmetry of any order around its centre. The circle is invariant under rotation by any angle about its centre and under reflection across any diameter.

Chords and arcs display a number of consistent relationships. All circles are similar; any circle can be mapped onto another by a uniform scaling followed by a translation. Equal chords are equidistant from the centre, and conversely, chords that are equidistant from the centre are equal in length. A perpendicular from the centre to a chord bisects the chord, and the perpendicular bisector of a chord passes through the centre. The measure of an inscribed angle is half the measure of its intercepted arc, a relationship that leads to Thales’s theorem: an angle inscribed in a semicircle is a right angle. The angle between a tangent and a chord through the point of tangency equals the angle in the alternate segment. For a point outside a circle, the two tangent segments to the circle have equal lengths. The power of a point theorem relates the lengths of secants, tangents, and chords drawn from a point to a circle.

Inscribed and Circumscribed Circles

A circle is said to be circumscribed about a polygon if all vertices of the polygon lie on the circle; in this case, the polygon is inscribed in the circle. Every triangle has a unique circumscribed circle, called its circumcircle; the centre of this circle is the circumcentre, found at the intersection of the perpendicular bisectors of the sides. A polygon that has a circumscribed circle is called a cyclic polygon; all regular polygons are cyclic. A quadrilateral is cyclic if and only if its opposite angles sum to \( 180^\circ \).

A circle is inscribed in a polygon if it is tangent to each side of the polygon; the polygon is then said to be circumscribed about the circle. Every triangle has a unique inscribed circle, its incircle. The centre is the incentre, the intersection of the angle bisectors. The radii of the circumcircle and incircle of a triangle are related to the triangle’s area, sides, and semiperimeter. For regular polygons, the incircle and circumcircle are concentric. The existence of inscribed and circumscribed circles for a given polygon leads to many of the classical constructions and theorems in planar geometry.

Theorems Involving Circles

Numerous classical theorems are built upon the circle:

• Thales’s theorem: If A, B, and C are points on a circle where AC is a diameter, then angle ABC is a right angle.
• Intersecting chords theorem: For two intersecting chords inside a circle, the products of the lengths of the segments of each chord are equal.
• Tangent–secant theorem: For a tangent and a secant from an external point, the square of the tangent length equals the product of the entire secant length and its external segment.
• Ptolemy’s theorem: For a cyclic quadrilateral, the product of the diagonals equals the sum of the products of opposite sides.
• Central angle theorem: The central angle subtended by an arc is twice any inscribed angle subtended by the same arc.
• Inscribed angle in a semicircle: An angle inscribed in a semicircle is right.

These theorems form the basis for many constructions, proofs, and problem-solving techniques in Euclidean geometry.

Area, Circumference, and Pi

The ratio of a circle's circumference to its diameter is constant and is denoted by the Greek letter \( \pi \). This constant is irrational and transcendental, approximately equal to 3.14159. Thus, if a circle has radius \( r \), its circumference \( C \) is given by:

\[
C = 2\pi r
\]

Its area \( A \) is given by:

\[
A = \pi r^2
\]

The area can be understood as the limit of the areas of inscribed regular polygons as the number of sides increases. This method of exhaustion was used by Archimedes, who gave rigorous bounds for \( \pi \). The area of a sector with central angle \( \theta \) (in radians) is \( \frac{1}{2} r^2 \theta \), and the length of the corresponding arc is \( r\theta \). For an annulus, the region between two concentric circles of radii \( R \) and \( r \), the area is \( \pi (R^2 - r^2) \).

Generalizations and Related Concepts

In non-Euclidean geometries, the definition of a circle adapts to the metric of the space. In spherical geometry, a circle on a sphere is the intersection of the sphere with a plane. If the plane passes through the centre of the sphere, the intersection is a great circle; otherwise, it is a small circle. Great circles are the geodesics of the sphere and play a role analogous to straight lines in the plane. In hyperbolic geometry, a circle is the set of points at a constant distance from a centre, and its circumference grows exponentially with the radius.

A generalised circle, also known as a cline or a circle in the inversive plane, includes both circles and straight lines, considered as circles passing through the point at infinity. Inversive geometry studies transformations that map generalised circles to generalised circles. In circle packing, arrangements of circles with specified tangency relations are studied for both theoretical and applied purposes. Apollonian gaskets are fractal sets generated by repeatedly filling interstices between mutually tangent circles. The circle is also the one-dimensional sphere, usually denoted \( S^1 \), and generalises to the higher-dimensional hyperspheres. In topology, the circle is a compact one-dimensional manifold without boundary. In complex analysis, the unit circle plays a central role as the boundary of the open unit disc, deeply connected to Fourier series and holomorphic functions.

Circles appear in numerous other sciences and engineering disciplines, including circular motion, optics, antenna theory, architecture, and art, where their symmetry and structural properties are continually exploited.