Circle

Circle

In the language of Euclidean geometry, a circle is the locus of points in a plane that are all at an equal distance from a single point, called the center of the circle. The fixed distance is called the radius of the circle. A line segment with each of its endpoints on the circle, which passes through the center of the circle, is called a diameter of the circle. The length of a diameter is twice the radius.

The distance around a circle, called its circumference, is the length of the line segment that would result if the circle were broken at a point and straightened out. This length is given by 2πr, where r is the radius of the circle and π (the Greek letter pi, pronounced “pie”) is a mathematical constant equal to approximately 3.14159. The circle’s arc is any portion of the circle lying between two points on the circle.

The history of the circle began before recorded history. Some historians think that the circle was likely first considered after the wheel was first invented. One of the first descriptions of the circle goes back to Egyptian scribe A hmes (1680–1620 BC) when he proposed a rule for finding the area of a circle. Later, Greek philosopher Thales of Miletus (624–547 BC) developed the first recorded theorems involving the properties of circles.

Points lying outside the circle are those points whose distance from the center is greater than the radius of the circle, and points lying in the circle are those points whose distance from the center is less than the radius of the circle. Any line that intersects the circumference at one point is tangent to the circle, while if the line intersects it at two points, then it is called a secant. The area covered by a circle, including all the points within it, is called the area of the circle. The area of a circle is also related to its radius by the formula A = πr², where A is the area, r is the radius, and π is the same constant as that in the formula for the circumference. Concentric circles are all circles that possess the same center but different lengths for diameters.

In the language of algebra, a circle corresponds to the set of ordered pairs (x, y) such that (x − a)² + (y − b)² = r², where the point corresponding to the ordered pair (a, b) is the center of the circle and the radius is equal to r. Based on this equation, circles are classified within the family of curves called conic sections. As such, circles are expressed as the intersection of a right circular cone with a plane that is perpendicular to the cone’s axis.