If an object does not have a center, the term may refer to itscircumradius, the radius of itscircumscribed circle orcircumscribed sphere. In either case, the radius may be more than half the diameter, which is usually defined as the maximum distance between any two points of the figure. Theinradius of a geometric figure is usually the radius of the largest circle or sphere contained in it. The inner radius of a ring, tube or other hollow object is the radius of its cavity.
Forregular polygons, the radius is the same as its circumradius.[4] The inradius of a regular polygon is also calledapothem. Ingraph theory, theradius of a graph is the minimum over all verticesu of the maximum distance fromu to any other vertex of the graph.[5]
The radius of the circle that passes through the three non-collinear pointsP1,P2, andP3 is given by
whereθ is the angle∠P1P2P3. This formula uses thelaw of sines. If the three points are given by their coordinates(x1,y1),(x2,y2), and(x3,y3), the radius can be expressed as
The radiusr of a regular polygon withn sides of lengths is given byr =Rns, where Values ofRn for small values ofn are given in the table. Ifs = 1 then these values are also the radii of the corresponding regular polygons.
The fixed point (analogous to the origin of aCartesian system) is called thepole, and theray from the pole in the fixed direction is thepolar axis. The distance from the pole is called theradial coordinate orradius, and the angle is theangular coordinate,polar angle, orazimuth.[6]
In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. Theorigin of the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis.
The axis is variously called thecylindrical orlongitudinal axis, to differentiate it fromthepolar axis, which is theray that lies in the reference plane, starting at the origin and pointing in the reference direction.
The distance from the axis may be called theradial distance orradius, while the angular coordinate is sometimes referred to as theangular position or as theazimuth.The radius and the azimuth are together called thepolar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane through the point, parallel to the reference plane.The third coordinate may be called theheight oraltitude (if the reference plane is considered horizontal),longitudinal position,[7] oraxial position.[8]
In a spherical coordinate system, the radius describes the distance of a point from a fixed origin. Its position if further defined by the polar angle measured between the radial direction and a fixed zenith direction, and the azimuth angle, the angle between the orthogonal projection of the radial direction on a reference plane that passes through the origin and is orthogonal to the zenith, and a fixed reference direction in that plane.
^Jonathan L. Gross, Jay Yellen (2006),Graph theory and its applications. 2nd edition, 779 pages; CRC Press.ISBN1-58488-505-X, 9781584885054.Online version accessed on 2009-08-08.
