![Circle-withsegments](https://upload.wikimedia.org/wikipedia/commons/0/03/Circle-withsegments.svg)
In classical
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a radius (
: radii) of a
circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
or
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
is any of the
line segments from its
center
Center or centre may refer to:
Mathematics
*Center (geometry), the middle of an object
* Center (algebra), used in various contexts
** Center (group theory)
** Center (ring theory)
* Graph center, the set of all vertices of minimum eccentrici ...
to its
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
, and in more modern usage, it is also their length. The name comes from the
latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
''radius'', meaning ray but also the spoke of a chariot wheel.
[Definition of Radius](_blank)
at dictionary.reference.com. Accessed on 2009-08-08. The plural of radius can be either ''radii'' (from the Latin plural) or the conventional English plural ''radiuses''. The typical abbreviation and
mathematical variable
In mathematics, a variable (from Latin language, Latin ''wikt:variabilis, variabilis'', "changeable") is a Mathematical symbol, symbol that represents a mathematical object. A variable may represent a number, a Vector (mathematics), vector, a Mat ...
name for radius is R or r. By extension, the
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
D is defined as twice the radius:
[Definition of radius]
at mathwords.com. Accessed on 2009-08-08.
:
If an object does not have a center, the term may refer to its circumradius, the radius of its
circumscribed circle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius.
Not every polyg ...
or
circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
. 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. The
inradius
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter.
...
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.
For
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
s, the radius is the same as its circumradius.
[Barnett Rich, Christopher Thomas (2008), ''Schaum's Outline of Geometry'', 4th edition, 326 pages. McGraw-Hill Professional. , ]
Online version
accessed on 2009-08-08. The inradius of a regular polygon is also called
apothem
The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides. T ...
. In
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, the
radius of a graph is the minimum over all vertices ''u'' of the maximum distance from ''u'' to any other vertex of the graph.
[Jonathan L. Gross, Jay Yellen (2006), ''Graph theory and its applications''. 2nd edition, 779 pages; CRC Press. , 9781584885054]
Online version
accessed on 2009-08-08.
The radius of the circle with
perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimeter has several pr ...
(
circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out t ...
) ''C'' is
:
Formula
For many geometric figures, the radius has a well-defined relationship with other measures of the figure.
Circles
The radius of a circle with
area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an ope ...
is
:
The radius of the circle that passes through the three non-
collinear
In geometry, collinearity of a set of points is the property of their lying on a single line. A set of points with this property is said to be collinear (sometimes spelled as colinear). In greater generality, the term has been used for aligned o ...
points , , and is given by
:
where is the angle . This formula uses the
law of sines
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law,
\frac \,=\, \frac \,=\, \frac \,=\, 2R,
where , and ar ...
. If the three points are given by their coordinates , , and , the radius can be expressed as
:
Regular polygons
![Regular polygon radius square](https://upload.wikimedia.org/wikipedia/commons/3/39/Regular_polygon_radius_square.png)
The radius of a regular polygon with sides of length is given by , where
Values of for small values of are given in the table. If then these values are also the radii of the corresponding regular polygons.
Hypercubes
The radius of a ''d''-dimensional
hypercube with side ''s'' is
:
Use in coordinate systems
Polar coordinates
The polar coordinate system is a
two
2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultur ...
-
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al
coordinate system in which each
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
on a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
is determined by a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from a fixed point and an
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
from a fixed direction.
The fixed point (analogous to the origin of a
Cartesian system) is called the ''pole'', and the
ray from the pole in the fixed direction is the ''polar axis''. The distance from the pole is called the ''radial coordinate'' or ''radius'', and the angle is the ''angular coordinate'', ''polar angle'', or ''
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
''.
Cylindrical coordinates
In the cylindrical coordinate system, there is a chosen reference axis and a chosen reference plane perpendicular to that axis. The ''origin'' 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 the ''cylindrical'' or ''longitudinal'' axis, to differentiate it from
the ''polar axis'', which is the
ray that lies in the reference plane,
starting at the origin and pointing in the reference direction.
The distance from the axis may be called the ''radial distance'' or ''radius'',
while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''.
The radius and the azimuth are together called the ''polar 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 the ''height'' or ''altitude'' (if the reference plane is considered horizontal),
''longitudinal position'',
or ''axial position''.
[ " ../nowiki>where ''r'', ''θ'', and ''z'' are cylindrical coordinates ../nowiki> as a function of axial position ../nowiki>"]
Spherical coordinates
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.
See also
*
Bend radius Bend radius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube, sheet, cable or hose without kinking it, damaging it, or shortening its life. The ''smaller'' the bend radius, the ''greater'' the material ...
*
Filling radius In Riemannian geometry, the filling radius of a Riemannian manifold ''X'' is a metric invariant of ''X''. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly genera ...
in Riemannian geometry
*
Radius of convergence
In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series ...
*
Radius of convexity
*
Radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
*
Radius of gyration ''Radius of gyration'' or gyradius of a body about the axis of rotation is defined as the radial distance to a point which would have a moment of inertia the same as the body's actual distribution of mass, if the total mass of the body were concentr ...
*
Semidiameter
References
{{Authority control
Spheres
Circles
Length