circle
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 (British English) or center (American English). 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 perimeter of a circle is also known as the circumference, 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
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, thus guaranteeing that all tangents are perpendicular to the radius and diameter that stem from the corresponding contact point on the circumference. 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 etymology of the word circle is from the Greek, kirkos "a circle," from the base ker- which means to turn or bend. The origin of the word "circus" is closely related as well.
The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle 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, astronomy, 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
.[1]
- 300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
- 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.
- 1880 – Lindemann proves that
Length of circumference
The ratio of a circle's circumference to its diameter is π (pi), an irrational 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
or equivalently to the diameter (d) by