Circle

Area of the circle = <math>\pi</math> × area of the shaded square

A circle is the set of all the points in a given plane that are the same distance, called the radius, r, from a given point called the center.

A circle may also be defined algebraically, as the set of solutions to an equation of the form:

<math>(x-a)^2 + (y-b)^2 = r^2</math>,

where <math>(a,b)</math> is the center of the circle. It can also be described parametrically in terms of a parameter <math>t</math> as:

<math>x = a + r \cos{t}</math>

<math>y = b + r \sin{t}</math>

The distance around the circle, or circumference, is given by:

<math>C = 2\pi r</math>

The area inside the circle is calculated using the formula:

<math>A = \pi r^2</math>

This can be easily derived from the area of an ellipse. For an ellipse with major and minor axis <math>a</math> and <math>b</math> respectively, the area is <math>\pi ab</math>. Setting the major an minor axis equal to each other give the formula for a circle.

A circle is a conic section, the intersection of a plane with a cone such that the plane is perpendicular to the axis of the cone.

Circles can readily be constructed by using a fixed distance between a pencil point and the center. A compass is a tool for doing this easily.

A circle with <math>r = 1</math> is called the unit circle, and is used extensively in trigonometry.