In classical geometry, a radius 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 name comes from the
Latin
d ≐ 2 r ⇒ r = d 2 . displaystyle ddoteq 2rquad Rightarrow quad r= frac d 2 . 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.[4] The inradius of a regular polygon is also called apothem. 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.[5] The radius of the circle with perimeter (circumference) C is r = C 2 π . displaystyle r= frac C 2pi . Contents 1 Formula 1.1 Circles 1.2 Regular polygons 1.3 Hypercubes 2 Use in coordinate systems 2.1 Polar coordinates 2.2 Cylindrical coordinates 2.3 Spherical coordinates 3 See also 4 References 5 External links Formula[edit] For many geometrical figures, the radius has a welldefined relationship with other measures of the figure. Circles[edit] The radius of a circle with area A is r = A π . displaystyle r= sqrt frac A pi . The radius of the circle that passes through the three noncollinear points P1, P2 and P3 is given by r =
O P 1 → − O P 3 →
2 sin θ , displaystyle r= frac vec OP_ 1  vec OP_ 3 2sin theta , where θ is the angle ∠ P 1 P 2 P 3 . displaystyle angle P_ 1 P_ 2 P_ 3 . This formula uses the law of sines. If the three points are given by their coordinates ( x 1 , y 1 ) displaystyle (x_ 1 ,y_ 1 ) , ( x 2 , y 2 ) displaystyle (x_ 2 ,y_ 2 ) and ( x 3 , y 3 ) displaystyle (x_ 3 ,y_ 3 ) , the radius can be expressed as r = ( ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 ) ( ( x 2 − x 3 ) 2 + ( y 2 − y 3 ) 2 ) ( ( x 3 − x 1 ) 2 + ( y 3 − y 1 ) 2 ) 2
x 1 y 2 + x 2 y 3 + x 3 y 1 − x 1 y 3 − x 2 y 1 − x 3 y 2
. displaystyle r= frac sqrt ((x_ 2 x_ 1 )^ 2 +(y_ 2 y_ 1 )^ 2 )((x_ 2 x_ 3 )^ 2 +(y_ 2 y_ 3 )^ 2 )((x_ 3 x_ 1 )^ 2 +(y_ 3 y_ 1 )^ 2 ) 2x_ 1 y_ 2 +x_ 2 y_ 3 +x_ 3 y_ 1 x_ 1 y_ 3 x_ 2 y_ 1 x_ 3 y_ 2 . Regular polygons[edit] See also: Circumscribed circle The radius of a regular polygon with n sides of length s is given by r = R n s displaystyle r=R_ n ,s , with R n = 1 / ( 2 sin π n ) displaystyle R_ n =1/left(2sin frac pi n right) and s = 1 : displaystyle s=1: n R n n R n 2 0.50000000 10 1.6180340 − 3 0.5773503 − 11 1.7747328 − 4 0.7071068 − 12 1.9318517 − 5 0.8506508 + 13 2.0892907 + 6 1.00000000 14 2.2469796 + 7 1.1523824 + 15 2.4048672 − 8 1.3065630 − 16 2.5629154 + 9 1.4619022 + 17 2.7210956 − displaystyle begin array rccrc n&R_ n &&n&R_ n \hline 2&0.50000000&&10&1.6180340\3&0.5773503&&11&1.7747328\4&0.7071068&&12&1.9318517\5&0.8506508+&&13&2.0892907+\6&1.00000000&&14&2.2469796+\7&1.1523824+&&15&2.4048672\8&1.3065630&&16&2.5629154+\9&1.4619022+&&17&2.7210956end array Hypercubes[edit] The radius of a ddimensional hypercube with side s is r = s 2 d . displaystyle r= frac s 2 sqrt d . Use in coordinate systems[edit] Polar coordinates[edit] Main article: Polar coordinate system The polar coordinate system is a twodimensional 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.[6] Cylindrical coordinates[edit] Main article: 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 twodimensional 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,[7] or axial position.[8] Spherical coordinates[edit] Main article: 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, that is, 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[edit] Bend radius
Filling radius in Riemannian geometry
Radius
References[edit] ^ Definition of
Radius
External links[edit]
Radius
