Circle

Circle, a well-known curve defined by Euclid as a plane figure bounded by a line such that every point on the line is equidistant from a certain point within the figure. This point is the centre, the boundary is called the circumference, and any line from centre to circumference is called a radius. All chords through the centre are therefore equal to twice the radius and are of equal length; such chords are called diameters, any one of which is an axis of symmetry of the figure. Other chords are equal if they lie at the same perpendicular distance from the centre; the length of a chord diminishes as its distance from the centre increases. The length of the circumference is a constant multiple of the diameter for all circles. Archimedes showed that this multiple, which is generally denoted by the Greek letter pi, was nearly 22/7. Other fractional approximations have been supplied, the neatest and best remembered being that of Metius 355/113. The exact multiple is incommensurable, that is, it cannot be expressed in a finite number of figures. It follows that all attempts to "square the circle," i.e. find a square of area exactly equal to a given circle, must be fruitless. It has been calculated to over 700 figures of decimals, starting thus - 3.14159265358979. Metius' fraction is correct to six decimal places. The circle is the only plane curve with the same curvature at every point. The curvature at every point in a straight line is the same, but is zero; in this sense we understand a straight line to be a special circle of infinite radius. Of all plane figures with the same circumference or perimeter, the circle has the greatest area. This area is pi times the square of the radius.

The circle is one of the conic sections, being obtained by cutting a right circular cone perpendicularly to its axis. In fact it may be treated as a special case of ellipse, with the two foci coincident, the directrices at infinity, and the two axes equal. As a conic, therefore, it can be cut by a straight line in two points only, and from any point only two tangents can be drawn to it; these two tangents are equal to each other.

The circular measure of an angle (q.v.) is of great importance in theory. Any angle at the centre of a circle of unit radius subtends a circular arc, whose length is proportional to the angle, and is therefore a measure of it.