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Radius
Radius
Radius
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Circle with:
  radius R
  circumference C
  diameter D
  center or origin O

In classical geometry, a radius (pl.: radii or radiuses)[a] 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]

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.[2] The inradius of a regular polygon is also called the apothem.[5] 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.[6]

The radius of the circle with perimeter (circumference) C is [b]


Formula

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For many geometric figures, the radius has a well-defined relationship with other measures of the figure.

Circles

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See also: Area of a circle

The radius of a circle with area A is



The radius of the circle that passes through the three non-collinear points P1, P2, and P3 is given by



where θ is the angle ∠P1P2P3. This formula uses the law of sines. If the three points are given by their coordinates (x1,y1), (x2,y2), and (x3,y3), the radius can be expressed as


Regular polygons

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See also: Circumscribed circle
n Rn
3 0.577350...
4 0.707106...
5 0.850650...
6
1
7 1.152382...
8 1.306562...
9 1.461902...
10 1.618033...
A square, for example (n=4)

The radius r of a regular polygon with n sides of length s is given by r = Rn s, where Values of Rn for small values of n are given in the table. If s = 1 then these values are also the radii of the corresponding regular polygons.


Hypercubes

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The radius of a d-dimensional hypercube with side s is

Use in coordinate systems

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Polar coordinates

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Main article: 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

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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 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

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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, 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

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Notes

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Radius

View on Grokipedia
from Grokipedia
In classical geometry, a radius (pl.: radii) of a circle or sphere is the length of a line segment from its center to any point on the perimeter. It is equal to half the diameter and serves as a fundamental measure for the size of circular and spherical objects. The concept of the radius is central to many areas of , including formulas for , area, and , as well as coordinate systems and applications in physics and .

Fundamentals

Definition

In , the radius refers to the from the center of a to any point on its . This is constant for all points on the , defining its size and shape as the set of all such equidistant points in a plane from the center. The term "radius" can also denote the itself connecting the center to a boundary point. For a in , the is analogously defined as the from to any point on the surface, with comprising all points equidistant from at this fixed length. This measure determines the sphere's and surface area, and the radius segment extends from the interior center to the boundary. In both cases, the radius is half the length of the , which passes through the center and connects two opposite boundary points. This fundamental concept underpins many geometric and calculations, such as for circles (2πr) and surface area for spheres (4πr²), where r denotes the radius.

Etymology and History

The term radius derives from the Latin noun radius, signifying a "ray," "beam of light," "spoke of a wheel," or "rod," with its geometric application emerging from the idea of a straight line extending from a like a ray. Its is *wreh₁dʰ- "to turn, twist," reflecting the spoke-like . This Latin word entered English around the 1590s, initially denoting a straight rod or beam, and by the 1610s, it specifically referred to the from the of a to its . The underlying concept of the radius—as the fixed distance from to the boundary of —originated in , where it was integral to understanding circular figures without a dedicated single term equivalent to the modern radius. In Euclid's Elements (circa 300 BCE), Book I defines as "a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another," establishing and the equal lengths (now called radii) as foundational, though described descriptively rather than named. The Latin radius first appeared as a technical geometrical term in Roman literature during the late , notably in the works of (106–43 BCE), who employed it to translate and adapt Greek ideas in discussions of astronomy and , such as in his of Plato's Timaeus where it denotes the spoke-like line in celestial circles. This usage marked an early terminological shift, bridging Greek conceptual foundations with Latin nomenclature during the Hellenistic influence on Roman scholarship. By the medieval period, mathematicians preserved and expanded these ideas through translations of , such as by in the early 9th century, indirectly influencing the term's persistence, though they favored descriptive phrases over direct adoption of radius. During the Renaissance, renewed interest in classical texts solidified radius in European mathematics, with its English geometric sense firmly established by the early . referenced it in 1656 as "the radius that describes the inner or outer circle," reflecting its integration into coordinate and amid the . This evolution underscored the term's transition from metaphorical "ray" to precise mathematical construct, enabling advancements in and physics where radial distances became central to modeling orbits and forces.

Formulas in Geometry

Circles and Spheres

In the geometry of circles, the radius rr serves as the fundamental distance from to any point on the , defining the circle as the locus of all points equidistant from . The CC, or perimeter, of a circle is given by the C=2πrC = 2\pi r, where π\pi is the approximately equal to 3.14159. This arises from the parameterization and the of the circle. The area AA enclosed by a circle is A=πr2A = \pi r^2, which can be derived through integration or geometric dissection into sectors. In Cartesian coordinates, the standard equation of a circle centered at (h,k)(h, k) with radius rr is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2, representing all points satisfying this distance condition. For spheres, the radius rr similarly defines the three-dimensional analog of a circle, as the set of all points in Euclidean space at a fixed distance from the center. The surface area SS of a sphere is S=4πr2S = 4\pi r^2, reflecting the sphere's uniformity across all great circles on its surface. This formula can be obtained by integrating the circumference of infinitesimal latitude circles or using Archimedean projections. The volume VV of the solid sphere (or ball) is V=43πr3V = \frac{4}{3}\pi r^3, a result historically attributed to via the and modernly confirmed through triple integration in spherical coordinates. The equation of a sphere centered at (a,b,c)(a, b, c) in three-dimensional Cartesian coordinates is (x−a)2+(y−b)2+(z−c)2=r2(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2, generalizing the circle's equation to higher dimensions. These formulas highlight the radius's central role in scaling geometric properties: linear dimensions like circumference scale with rr, surface areas with r2r^2, and volumes with r3r^3, illustrating dimensional progression in Euclidean geometry.

Regular Polygons and Polyhedra

In regular polygons, the radius manifests in two primary forms: the circumradius RR, which is the radius of the circle passing through all vertices, and the inradius rr, also known as the apothem, which is the radius of the circle tangent to all sides. These radii are derived by dividing the polygon into nn congruent isosceles triangles from the center, where nn is the number of sides. The central angle of each triangle is 2π/n2\pi/n, leading to the relation R=s2sin⁡(π/n)R = \frac{s}{2 \sin(\pi/n)} for side length ss, and r=s2tan⁡(π/n)r = \frac{s}{2 \tan(\pi/n)}. Alternatively, r=Rcos⁡(π/n)r = R \cos(\pi/n), highlighting the geometric connection between the two radii. For example, in an (n=3n=3), the circumradius is R=s3R = \frac{s}{\sqrt{3}}
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