This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.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 bounding line is called its circumference and the point, its centre.

A circle may also be defined as a special kind of ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal.

A circle is a simple closed curve which divides 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, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a disc.

Angles inscribed on the arc (brown) are supplementary.

In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Since the diameter is twice the radius, the "missing" part of the diameter is (. There are many compass-and-straightedge constructions resulting in circles.

The simplest and most basic is the construction given the centre of the circle and a point on the circle.

In polar coordinates the equation of a circle is: is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). r An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red).

Hence, all inscribed angles that subtend the same arc (pink) are equal.

In homogeneous coordinates each conic section with the equation of a circle has the form 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).