The length of a line segment connecting two points on the circle and passing through the centre is called the diameter.

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.

The Egyptian Rhind papyrus, dated to 1700 BCE, gives a method to find the area of a circle.

The bounding line is called its circumference and the point, its centre.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.

[7] With the advent of abstract art in the early 20th century, geometric objects became an artistic subject in their own right.

[8][9] 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 ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654.

The circle is the plane curve enclosing the maximum area for a given arc length.

and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is

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|.

The equation can be written in parametric form using the trigonometric functions sine and cosine as

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.

In this parameterisation, the ratio of t to r 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).

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).

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.)

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane.

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

The circle can be viewed as a limiting case of various other figures: Consider a finite set of

In the case of the equilateral triangle, the loci of the constant sums of the second and fourth powers are circles, whereas for the square, the loci are circles for the constant sums of the second, fourth, and sixth powers.

For the regular pentagon the constant sum of the eighth powers of the distances will be added and so forth.

Taxicab circles are squares with sides oriented at a 45° angle to the coordinate axes.