Circle
The set of all points in a plane that are equidistant from a fixed point, defined as the center, is called a circle.
![](/img/a/geometry/circles/circles/circle.png)
The circle above is called circle O. A line segment with two endpoints, one on the center and the other on the perimeter of the circle, forms a radius, r, of the circle.
![](/img/a/geometry/circles/circles/circle-and-lines.png)
A chord is a line segment whose endpoints lie on the perimeter of the circle. If the chord contains the center, it is called a diameter of the circle. The diameter is twice the length of the radius.
A line is called a secant line if it intersects the circle at two points. A line is called a tangent line if it intersects the circle at only one point, called a point of tangency.
Circles are widely studied in geometry. The following are some important concepts related to circles.
Circumference
The circumference, C, of a circle is similar to the perimeter of a polygon in that it is a measure of the distance around the circle.
![](/img/a/geometry/circles/circles/circumference.png)
The circumference of circle above can be found using the formula
C = 2πr
where r is the circle's radius. You can also use the formula C = πd, where d is the diameter of the circle. π is a mathematical constant that represents the ratio of the circle's circumference to its diameter. It is approximately equal to 3.14.
Example:
The figure below shows circle A with radius AB = 4.4 that has a circumference
C = 2π·4.4 = 27.6.
![](/img/a/geometry/circles/circles/circumference-example.png)
The line segment below the circle represents the straight length of its circumference.
Arc of a circle
An arc of a circle is part of a circle.
![](/img/a/geometry/circles/circles/arc.png)
Arc AB, written symbolically as , is shown in red for circle O.
is called a minor arc since it encompasses less than one-half the circles circumference.
is called a major arc since it encompasses more than one-half the circle's circumference. Minor arcs are usually named using two points while major arcs are usually named using three.
An arc's length is a portion of the length of the circumference of a circle. It can be calculated by using the formula,
where θ is the central angle, in units of degrees, subtended by the arc. If the angle is given in radians, the arc length can be found using:
s = rθ
For circle O below, the measure of is,
![](/img/a/geometry/circles/circles/arc-example.png)
Area
The area of a circle is the plane region bounded by the circle's circumference. Area is measured in square units of length while circumference is measured units of length.
![](/img/a/geometry/circles/circles/area.png)
The region shaded in gray inside of circle O above is its area. The formula for the area is
A = πr2
where r is the circle's radius.
Sector
A sector of a circle is the enclosed region bounded by two radii and the arc they form. Sector OAB below, formed by radii OA, OB, and , is the gray region of circle O.
![](/img/a/geometry/circles/circles/sector.png)
The area, A, of a sector can be calculated by using the formula,
where θ is the central angle (in degrees) that forms the sector. If the angle is given in radians, the area of the sector can be found using:
For circle O below, the area of sector OAB is,
![](/img/a/geometry/circles/circles/sector-example.png)
Alternatively, the area of a sector can be found using the formula,
where r is the radius and s is the sector's arc length.
Did you know?
The region between two concentric circles is called an annulus. The name comes from a Latin root meaning "little ring." The term "annulus" is also used to name ring-like structures such as the yearly growth ring of a tree: