Circle (1)

A circle is a shape that can be found in any plane, and is defined by all points that are a certain distance away from a central point. This distance is known as the radius. It is important to note that the radius is typically a positive number. A circle can also be thought of as a simple, closed curve that separates a plane into an interior and an exterior region. In common usage, the term "circle" may refer to either the boundary of the shape or the entire shape, including the interior region. The boundary of the shape is technically just the circle, while the entire shape is called a disc.

Another way to define a circle is as a type of ellipse in which the two foci are at the same point, the eccentricity is 0, and the semi-major and semi-minor axes are equal. Alternatively, a circle can be seen as the two-dimensional shape that encloses the most area per unit of perimeter squared, according to the principles of calculus of variations.

A circle is a geometric figure bounded by a single curved line, such that all straight lines drawn from a specific point within it to the bounding line are equal in length. This bounding line is known as the circumference of the circle, and the specific point is called the center. In the field of topology, a circle is not limited to the geometric definition, but also includes all of its homeomorphisms. Two topological circles are considered equivalent if one can be transformed into the other through a deformation of three-dimensional space upon itself (also known as an ambient isotopy).

The following are various aspects of circles:

Annulus: a ring-shaped object, defined by two concentric circles.

Arc: any connected part of a circle. By specifying the two endpoints of an arc and the center, we can determine two arcs that together form a full circle.

Center: the point equidistant from all points on the circle.

Chord: a line segment whose endpoints lie on the circle, dividing the 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 pass through the center, or the length of such a line segment. This is the largest distance between any two points on the circle. It is a specific type of chord, namely the longest chord for a given circle, and its length is twice the length of the radius.

Disc: the region of the plane bounded by a circle.

Lens: the region common to two overlapping discs.

Passant: a straight line that is coplanar with the circle and has no points in common with it.

Radius: a line segment connecting the center of a circle to any single point on the circle itself, or the length of such a segment, which is half the length of the diameter.

Sector: a region bounded by two radii of equal length with a common center and one of the two possible arcs determined by this center and the endpoints of the radii.

Segment: a region bounded by a chord and one of the arcs connecting the endpoints of the chord. The length of the chord imposes a lower boundary on the diameter of possible arcs. Sometimes the term segment is used only for regions that do not contain the center of the circle to which their arc belongs.

Secant: an extended chord, a straight line that is coplanar with the circle and intersects it at two points.

Semicircle: one of the two possible arcs determined by the endpoints of a diameter, with the midpoint of the diameter as the center. In non-technical language, it may refer to the interior of the two-dimensional region bounded by a diameter and one of its arcs, which is technically known as a half-disc. A half-disc is a specific type of segment, namely the largest one.

Tangent: a straight line that is coplanar with the circle and touches it at a single point.

All of the aforementioned regions can be considered either open (not including their boundaries) or closed (including their respective boundaries).

The word "circle" has its roots in the Greek language, with "kirkos" and "kuklos" being the original terms for a hoop or ring. These words eventually evolved into the modern English term "circle" through a process known as metathesis. The concept of a circle has been present since ancient times, and has been observed in various natural phenomena such as the movement of the Sun and Moon, as well as the circular shape formed by a plant stalk blowing in the wind on sand. The circle is an important geometric shape that has inspired the development of various fields of study, including geometry, astronomy, and calculus.

In the history of mathematics and science, the circle has held a special place due to its perceived divine or perfect qualities. In ancient times, scholars believed that the circle was connected to the divine, and early advancements in fields such as geometry and astronomy were often linked to the divine. The study of the circle has led to numerous discoveries and innovations over the centuries, such as the method for finding the area of a circular field outlined in the Rhind papyrus around 1700 BCE, and the proof of the transcendental nature of the mathematical constant π by Lindemann in 1880 CE.

The circumference of a circle is related to its radius and diameter by the mathematical constant pi (π). Specifically, the circumference is equal to 2π times the radius, or π times the diameter. The area enclosed by a circle can also be calculated using pi, and is equal to π times the radius squared. Alternatively, the area can be expressed as approximately 79% of the area of the square that circumscribes the circle (that is, a square with sides of the same length as the diameter of the circle).

It is worth noting that the circle encloses the maximum possible area for a given arc length, which is a principle related to the calculus of variations and the isoperimetric inequality.