From en.wikipedia.org:
[Segment in a circle or sphere from its center to its perimeter or surface and
its length] [the line segment]
[[File:Circle-withsegments.svg|thumb|right|Circle with: [red solid 2px] [black
solid 3px] [blue solid 2px] [green solid 2px]]]
In classical geometry, a RADIUS ([plural form]: RADII or RADIUSES){{efn|The
plural of radius can be either _radii_ (from the Latin plural) or the
conventional English plural _radiuses_.[1]}} of a circle or sphere is any of
the line segments from its center to its perimeter, and in more modern usage,
it is also their length. The radius of a regular polygon is the line segment
or distance from its center to any of its vertices.[2] The name comes from the
Latin _radius_, meaning ray but also the spoke of a chariot wheel.[3] The
typical abbreviation and mathematical symbol for radius is _R_ or _r_. By
extension, the diameter _D_ is defined as twice the radius:[4]
<math> d \doteq 2r \quad \Rightarrow \quad r = \frac d 2. </math>
If an object does not have a center, the term may refer to its CIRCUMRADIUS,
the radius of its circumscribed circle or circumscribed 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. The inradius 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 polygons, the radius is the same as its circumradius.[5] The
inradius of a regular polygon is also called the apothem.[6] In graph theory,
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.[7]
The radius of the circle with perimeter (circumference) _C_ is
<math> r = \frac C {2\pi}. </math>
** Formula
For many geometric figures, the radius has a well-defined relationship with
other measures of the figure.
*** Circles
[Area of a circle]
The radius of a circle with area [_A_] is
<math> r = \sqrt{\frac{A}{\pi }} . </math>
The radius of the circle that passes through the three non-collinear points
[_P_ <sub> 1 </sub>], [_P_ <sub> 2 </sub>], and [_P_ <sub> 3 </sub>] is given
by
<math> r=\frac{ | \vec{OP_1}-\vec{OP_3} | }{2\sin\theta}, </math>
where [θ] is the angle [∠_P_ <sub> 1 </sub> _P_ <sub> 2 </sub> _P_ <sub> 3
</sub>]. This formula uses the law of sines. If the three points are given
by their coordinates [(_x_ <sub> 1 </sub> , _y_ <sub> 1 </sub> )], [(_x_ <sub>
2 </sub> , _y_ <sub> 2 </sub> )], and [(_x_ <sub> 3 </sub> , _y_ <sub> 3
</sub> )], the radius can be expressed as
<math> r = \frac {\sqrt{\bigl((x_2 - x_1)^2 + (y_2 - y_1)^2\bigr) \bigl((x_2 - x_3)^2 + (y_2 - y_3)^2\bigr) \bigl((x_3 - x_1)^2 + (y_3 - y_1)^2 \bigr)} }{ 2\bigl | x_1 y_2 + x_2 y_3 + x_3 y_1 - x_1 y_3 - x_2 y_1 - x_3 y_2\bigr | }. </math>
*** Regular polygons
[Circumscribed circle]
{| class="wikitable floatright" ! [n] ! [_R_ <sub> _n_ </sub>] |- | 3 ||
[0.577] |- | 4 || [0.707] |- | 5 || [0.850] |- | 6 || [1=1] |- | 7 || [1.152]
|- | 8 || [1.306] |- | 9 || [1.461] |- | 10 || [1.618] |}
The radius [r] of a regular polygon with [n] sides of length [s] is given by
[1= _r_ = _R_ <sub> _n_ </sub> _s_], where <math>R_n = 1\left/\left(2 \sin
\frac\pi n \right)\right. .</math> Values of [_R_ <sub> _n_ </sub>] for small
values of [n] are given in the table. If [1= _s_ = 1] then these values are
also the radii of the corresponding regular polygons.
<!-- To add: radius from area, inradius from outradius, outradius from
inradius -->
*** Hypercubes
The radius of a _d_-dimensional hypercube with side _s_ is
<math> r = \frac{s}{2}\sqrt{d}. </math>
** Use in coordinate systems
*** Polar coordinates
[Polar coordinate system] The polar coordinate system is a two-dimensional
coordinate system in which each point on a plane is determined by a distance
from a fixed point and an angle 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_.[8]
*** Cylindrical coordinates
[Cylindrical coordinate system] 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_,[9] or
_axial position_.[10]
*** Spherical coordinates
[Spherical coordinate system] 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
[colbegin]
- Bend radius
- Filling radius in Riemannian geometry
- Mean radius
- Radius of convergence
- Radius of convexity
- Radius of curvature
- Radius of gyration
- Semidiameter
[Div col end]
** Notes
[Notelist]
** References
[2]
[Authority control]
*