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Radius

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.^[1] It is equal to half the diameter and serves as a fundamental measure for the size of circular and spherical objects.^[2] The concept of the radius is central to many areas of mathematics, including formulas for circumference, area, and volume, as well as coordinate systems and applications in physics and engineering.^[3]

Fundamentals

Definition

In geometry, the radius refers to the distance from the center of a circle to any point on its circumference. This distance is constant for all points on the circle, 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 line segment itself connecting the center to a boundary point.^[4] For a sphere in three-dimensional space, the radius is analogously defined as the distance from the center to any point on the surface, with the sphere comprising all points equidistant from the center at this fixed length. This measure determines the sphere's volume and surface area, and the radius segment extends from the interior center to the boundary.^[5]^[6] In both cases, the radius is half the length of the diameter, which passes through the center and connects two opposite boundary points. This fundamental concept underpins many geometric properties and calculations, such as circumference for circles (2πr) and surface area for spheres (4πr²), where r denotes the radius.^[3]^[5]

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 center like a ray. Its Proto-Indo-European root is *wreh₁dʰ- "to turn, twist," reflecting the spoke-like nature. This Latin word entered English around the 1590s, initially denoting a straight rod or beam, and by the 1610s, it specifically referred to the line segment from the center of a circle to its circumference.^[7] The underlying concept of the radius—as the fixed distance from the center to the boundary of a circle or sphere—originated in ancient Greek geometry, 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 a circle 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 the center and the equal lengths (now called radii) as foundational, though described descriptively rather than named.^[8] The Latin radius first appeared as a technical geometrical term in Roman literature during the late Republic, notably in the works of Cicero (106–43 BCE), who employed it to translate and adapt Greek ideas in discussions of astronomy and geometry, such as in his translation of Plato's Timaeus where it denotes the spoke-like line in celestial circles.^[9] 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, Arabic mathematicians preserved and expanded these ideas through translations of Euclid, such as by al-Hajjaj ibn Yusuf in the early 9th century, indirectly influencing the term's persistence, though they favored descriptive Arabic phrases over direct adoption of radius.^[10] During the Renaissance, renewed interest in classical texts solidified radius in European mathematics, with its English geometric sense firmly established by the early 17th century. Thomas Hobbes referenced it in 1656 as "the radius that describes the inner or outer circle," reflecting its integration into coordinate and analytic geometry amid the scientific revolution.^[11] This evolution underscored the term's transition from metaphorical "ray" to precise mathematical construct, enabling advancements in calculus and physics where radial distances became central to modeling orbits and forces.^[11]

Formulas in Geometry

Circles and Spheres

In the geometry of circles, the radius r serves as the fundamental distance from the center to any point on the circumference, defining the circle as the locus of all points equidistant from the center.^[12] The circumference C, or perimeter, of a circle is given by the formula C = 2\pi r, where \pi is the mathematical constant approximately equal to 3.14159.^[12] This formula arises from the arc length parameterization and the rotational symmetry of the circle.^[12] The area A enclosed by a circle is A = \pi r^2, which can be derived through integration or geometric dissection into sectors.^[12] In Cartesian coordinates, the standard equation of a circle centered at (h, k) with radius r is (x - h)^2 + (y - k)^2 = r^2, representing all points satisfying this distance condition.^[12] For spheres, the radius r 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.^[5] The surface area S of a sphere is S = 4\pi r^2, reflecting the sphere's uniformity across all great circles on its surface.^[5] This formula can be obtained by integrating the circumference of infinitesimal latitude circles or using Archimedean projections.^[5] The volume V of the solid sphere (or ball) is V = \frac{4}{3}\pi r^3, a result historically attributed to Archimedes via the method of exhaustion and modernly confirmed through triple integration in spherical coordinates.^[5] The equation of a sphere centered at (a, b, c) in three-dimensional Cartesian coordinates is (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2, generalizing the circle's equation to higher dimensions.^[5] These formulas highlight the radius's central role in scaling geometric properties: linear dimensions like circumference scale with r, surface areas with r^2, and volumes with r^3, illustrating dimensional progression in Euclidean geometry.^[12]^[5]

Regular Polygons and Polyhedra

In regular polygons, the radius manifests in two primary forms: the circumradius R, which is the radius of the circle passing through all vertices, and the inradius r, 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 n congruent isosceles triangles from the center, where n is the number of sides. The central angle of each triangle is $2\pi/n, leading to the relation R = \frac{s}{2 \sin(\pi/n)} for side length s, and r = \frac{s}{2 \tan(\pi/n)}. Alternatively, r = R \cos(\pi/n), highlighting the geometric connection between the two radii.^[13] For example, in an equilateral triangle (n=3), the circumradius is R = \frac{s}{\sqrt{3}}, while the inradius is r = \frac{s \sqrt{3}}{6}, reflecting the height division where the inradius is one-third of the height. In a square (n=4), both radii simplify to R = \frac{s}{\sqrt{2}} and r = \frac{s}{2}, with the inradius being half the side length due to the right-angle symmetry. These formulas enable calculations of area as A = \frac{1}{2} n s r or A = \frac{1}{2} n s^2 \cot(\pi/n), emphasizing the radius's role in polygonal metrics.^[14] Extending to three dimensions, regular polyhedra, or Platonic solids, possess analogous inradius and circumradius, defined with respect to inscribed and circumscribed spheres relative to edge length a. The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron—each have unique expressions derived from their symmetry and face configurations. For the regular tetrahedron, the circumradius is R = \frac{\sqrt{6}}{4} a and the inradius is r = \frac{\sqrt{6}}{12} a, obtained by considering the distance from the centroid to a vertex and to a face center, respectively.^[15] The cube provides a simpler case, with circumradius R = \frac{\sqrt{3}}{2} a (distance to vertex from center) and inradius r = \frac{a}{2} (distance to face). For the regular octahedron, dual to the cube, R = \frac{\sqrt{2}}{2} a and r = \frac{\sqrt{6}}{6} a. The dodecahedron and icosahedron, with pentagonal and triangular faces, yield more complex forms: for the dodecahedron, R = \frac{\sqrt{3}}{4} (1 + \sqrt{5}) a and r = \frac{\sqrt{10 + 2\sqrt{5}}}{4} a; for the icosahedron, R = \frac{\sqrt{2}}{4} \sqrt{10 + 2\sqrt{5}} \, a and r = \frac{\sqrt{42 + 18\sqrt{5}}}{12} a. These radii facilitate volume computations, such as V = \frac{4}{3} \pi r^3 for the insphere, and underscore the solids' isoperimetric properties in packing and tiling applications.^[15] The ratio R/r, known as asphericity, measures deviation from sphericity, with values ranging from 3 for the tetrahedron to approximately 1.05 for the dodecahedron and icosahedron, influencing their densest packings.^[16]

Hypercubes and Higher Dimensions

In n-dimensional Euclidean space, the hypercube (or n-cube) with side length s is a regular polytope bounded by $2n pairwise perpendicular hyperplanes. The inradius r of an n-cube, defined as the radius of the inscribed hypersphere tangent to all faces, is r = s/2. This follows from the geometry of the hypercube, where the center lies at equal distance s/2 from each bounding hyperplane along the coordinate axes. The circumradius R of an n-cube, the radius of the circumscribed hypersphere passing through all $2^n vertices, is given by R = \frac{s \sqrt{n}}{2}. This distance arises as the Euclidean norm from the center to any vertex; for a hypercube centered at the origin with vertices at (\pm s/2, \pm s/2, \dots, \pm s/2), the vertex coordinates yield \sqrt{n (s/2)^2} = \frac{s \sqrt{n}}{2}. For the unit hypercube with s = 1, R = \sqrt{n}/2, which exceeds 1 for n > 4, illustrating how vertices increasingly lie farther from the center as dimensionality grows.^[17] These radii provide key measures for embedding hypercubes in higher-dimensional spaces and analyzing their metric properties. For instance, the ratio R/r = \sqrt{n} highlights the disparity between the inscribed and circumscribed hyperspheres, a characteristic that intensifies with dimension n. Seminal treatments of such polytopal metrics appear in foundational works on regular polytopes.

Role in Coordinate Systems

Polar and Cylindrical Coordinates

In polar coordinates, a point in the two-dimensional plane is specified by a pair (r, [\theta](/page/Theta)), where r represents the radial distance from the origin (the pole) to the point, and [\theta](/page/Theta) is the angular coordinate measured counterclockwise from the positive x-axis.^[18] The radius r is always non-negative and serves as the distance along a ray from the origin at angle [\theta](/page/Theta), enabling the representation of curves such as circles and spirals in a form that highlights rotational symmetry./11%3A_Parametric_Equations_and_Polar_Coordinates/11.03%3A_Polar_Coordinates) For instance, the polar equation r = k for a constant k > 0 describes a circle of radius k centered at the origin./11%3A_Parametric_Equations_and_Polar_Coordinates/11.03%3A_Polar_Coordinates) The conversion between polar and Cartesian coordinates underscores the role of the radius: x = r \cos \theta and y = r \sin \theta, with r = \sqrt{x^2 + y^2}.^[18] This formulation arises from the geometric interpretation of r as the hypotenuse in a right triangle formed by the point's projections on the axes, facilitating applications in integration and vector analysis where radial symmetry simplifies computations./11%3A_Parametric_Equations_and_Polar_Coordinates/11.03%3A_Polar_Coordinates) Cylindrical coordinates extend polar coordinates to three dimensions by incorporating a height coordinate z, representing a point as (r, \theta, z), where r and \theta define the projection onto the xy-plane exactly as in polar coordinates, and z measures the vertical distance from that plane.^[19] Here, the radius r retains its role as the perpendicular distance from the z-axis to the point, forming the basis for describing surfaces of revolution and cylindrical symmetries in space. The Cartesian equivalents are x = r \cos \theta, y = r \sin \theta, and z = z, with r = \sqrt{x^2 + y^2}, confirming r's invariance as the radial component in the horizontal plane.^[19] Surfaces defined by constant r = c (for c > 0) are vertical cylinders of radius c aligned with the z-axis, illustrating how the radius parameterizes infinite tubes in three-dimensional geometry. This system is particularly useful in physics for problems involving axial symmetry, such as electromagnetic fields around wires, where r directly influences potential and field strength variations.

Spherical Coordinates

In spherical coordinates, a point in three-dimensional Euclidean space is represented by the ordered triple (\rho, \theta, \phi), where \rho denotes the radial distance from the origin (also called the radius vector length), \theta is the azimuthal angle measured from the positive x-axis in the xy-plane, and \phi is the polar angle measured from the positive z-axis. The radial coordinate \rho satisfies \rho \geq 0, while \theta ranges from 0 to $2\pi and \phi ranges from 0 to \pi.^[20] This system extends the two-dimensional polar coordinates by incorporating a third dimension, with \rho serving as the generalization of the radial distance r.^[21] The role of the radius \rho is central, as it defines the distance from the origin to the point, enabling the description of spherical symmetry. The surfaces of constant \rho form spheres centered at the origin, with radius \rho. For instance, the equation \rho = a (where a > 0) describes a sphere of radius a.^[21] Constant \phi yields cones aligned with the z-axis, and constant \theta produces half-planes emanating from the z-axis.^[20] The transformation from spherical to Cartesian coordinates is given by:

\begin{align*} x &= \rho \sin \phi \cos \theta, \\ y &= \rho \sin \phi \sin \theta, \\ z &= \rho \cos \phi. \end{align*}

These equations highlight how \rho scales the position vector, with the angular components determining its direction.^[20] The inverse transformation, yielding \rho = \sqrt{x^2 + y^2 + z^2}, \theta = \atan2(y, x), and \phi = \arccos\left(\frac{z}{\rho}\right), directly expresses the radius as the Euclidean norm of the Cartesian position vector.^[22] In multivariable calculus, spherical coordinates are particularly useful for integrating over regions with spherical symmetry, where the Jacobian determinant introduces a factor of \rho^2 \sin \phi in the volume element dV = \rho^2 \sin \phi \, d\rho \, d\theta \, d\phi.^[23] This \rho^2 term arises from the scaling effect of the radial coordinate in three dimensions, facilitating computations for volumes, masses, or potentials inside spheres or around point sources. For example, the volume of a sphere of radius R is computed as \int_0^{2\pi} \int_0^\pi \int_0^R \rho^2 \sin \phi \, d\rho \, d\phi \, d\theta = \frac{4}{3} \pi R^3.^[20]

Applications

In Physics and Astronomy

In physics, the radius concept is fundamental to describing curved trajectories and spatial extents in various phenomena. In optics, the radius of curvature R of a spherical mirror or lens surface determines the focal length, given by f = R/2 for a concave mirror, which governs how light rays converge or diverge.^[24] This property is essential for designing optical instruments like telescopes and microscopes, where precise curvature radii ensure accurate image formation. Similarly, in electromagnetism, charged particles moving perpendicular to a uniform magnetic field follow circular paths with radius r = \frac{m v}{q B}, where m is mass, v is speed, q is charge, and B is magnetic field strength; this cyclotron radius enables applications in particle accelerators and mass spectrometers.^[25] In plasma physics, the Larmor radius r_L = \frac{m v_\perp}{q B}, with v_\perp as the velocity component perpendicular to the field, quantifies the gyroradius of ions and electrons in magnetized plasmas, setting the scale for confinement in fusion devices like tokamaks.^[26] For instance, in a 1 T field, electrons in a hydrogen plasma have a Larmor frequency of 28 GHz, leading to micrometer-scale radii that facilitate controlled energy transport. In general relativity, the Schwarzschild radius r_s = \frac{2 G M}{c^2}, where G is the gravitational constant, M is mass, and c is the speed of light, defines the event horizon of a non-rotating black hole, beyond which escape is impossible.^[27] For the Sun, this yields r_s \approx 2.95 km, illustrating the scale where spacetime curvature becomes extreme.^[27] In astronomy, radius measurements reveal the sizes of celestial bodies and orbits. Stellar radii are inferred from spectra, where line widths indicate atmospheric pressure: giant stars exhibit narrow lines due to low-density photospheres, implying radii hundreds of times the Sun's 696,000 km.^[28] Direct measurements via interferometry, such as with the CHARA array, confirm radii from 0.15 to 848 solar radii across main-sequence and evolved stars.^[29] Planetary radii, often expressed in Earth radii (R_\Earth \approx 6,371 km), are determined from transits; for example, the exoplanet Kepler-22b has a radius of approximately 2.1 R_\Earth (as of 2025), placing it in the super-Earth category within habitable zones.^[30] Orbital radii, approximated as the semi-major axis in Kepler's third law T^2 \propto a^3, describe planetary and satellite paths, as in Earth's 1 AU (\approx 149.6 million km) orbit, crucial for mission planning.^[31]

In Engineering and Other Fields

In mechanical engineering, the radius of gyration is a critical parameter used to characterize the distribution of a structural member's cross-sectional area relative to its centroidal axis, aiding in the prediction of buckling behavior under compressive loads. It is defined as the square root of the moment of inertia divided by the cross-sectional area, providing a measure of the member's resistance to torsional and flexural buckling; for instance, in column design, a larger radius of gyration indicates greater stability against Euler buckling. This concept is fundamental in structural analysis, as outlined in standard engineering references for beams and columns.^[32] Bend radius plays a vital role in manufacturing processes, particularly in sheet metal fabrication, where it specifies the minimum inside curvature a material can undergo without cracking or excessive springback, directly influencing part integrity and formability. For materials like aluminum or steel, the bend radius is typically expressed as a multiple of the material thickness—such as 1t for mild steel—to ensure ductility is maintained during bending operations like air bending or bottoming. In cable and conduit design, adhering to a minimum bend radius, often 4 to 10 times the cable diameter, prevents signal attenuation or mechanical failure in electrical installations.^[33]^[34] Fillet radius is employed in product design and machining to round off sharp internal corners, thereby reducing stress concentrations that could lead to fatigue failure in components under cyclic loading. By distributing loads more evenly, a fillet radius enhances the durability of parts such as gears, shafts, and pressure vessels; typical values range from 0.5 to 2 times the material thickness, depending on the application's stress analysis. This practice is standard in finite element modeling to simulate realistic boundary conditions and improve safety factors.^[35] In civil engineering, the radius of curvature defines the sharpness of horizontal alignments in roadways, railways, and bridges, ensuring safe vehicle dynamics by balancing centrifugal forces with superelevation. For highways, design standards recommend minimum radii—such as approximately 240 meters for curves at 80 km/h—to minimize lateral acceleration and enhance ride comfort, calculated using the formula R = \frac{V^2}{127(e + f)}, where V is speed, e is superelevation, and f is side friction. This parameter is integral to geometric design guidelines for infrastructure projects.^[36] In geometric dimensioning and tolerancing (GD&T), the radius symbol "R" precedes dimensional values to specify the exact curvature of features like arcs or fillets on engineering drawings, facilitating precise manufacturing and quality control. For example, a controlled radius ensures interchangeability in assemblies by tolerating variations within specified limits, as per ASME Y14.5 standards, preventing over- or under-specification in production.^[37]

References

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Anatomy, Shoulder and Upper Limb, Forearm Radius - NCBI - NIH
The radius is one of two long bones that make up the human antebrachium, the other bone being the ulna. The radius has three borders, three surfaces, and has a ...
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The radius is the lateral bone of the forearm. It is a long bone that has three main parts: a proximal end, shaft and a distal end.
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Feb 28, 2021 · The main parts of the proximal radius are the head, the neck and the radial tuberosity. Head: articulates with the capitulum of the humerus.
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Radius - Physiopedia
The radius is one of the two bones that make up the forearm, the other being the ulna. It forms the radio-carpal joint at the wrist and the radio-ulnar joint ...
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Radius: Definition, Function, Common Conditions & More
The radius is one of the two major bones in the forearm (the other is the ulna). The radius runs from the elbow to the wrist.
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Radius (Bone): Anatomy, Location & Function - Cleveland Clinic
The radius is one of the bones in your forearm. It helps you move your arm and wrist. Your radius also supports lots of important muscles, tendons, ligaments ...
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Nov 16, 2022 · A circle is all the points that are the same distance, r r – called the radius, from a point, (h,k) ( h , k ) - called the center. In other ...Missing: geometry | Show results with:geometry<|control11|><|separator|>
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Sphere - Definition With Examples We know that the diameter of a sphere is given by d = 2r, where r is the radius. For the given sphere, r = 25 cm. Therefore, ...
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From 1590s Latin radius, meaning "rod, spoke, or ray," origin uncertain but possibly linked to radix "root"; denotes a straight rod or beam of light.
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Earliest Known Uses of Some of the Words of Mathematics (R)
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pa=r^2. (12). The circle having P_1P_2 as a diameter is given by. (x-x_1)(x-x_2)+(y-y_1). (13). The parametric equations for a circle of radius a can be given ...
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[PDF] Chapter 7.2-7.3 Triangle Centers and Regular Polygon Properties
The inradius of a regular polygon is exactly the same as its apothem. The formulas below are the same as for the apothem. Circumcircle of a Polygon. Definition: ...
[18]
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[20]
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The polar coordinates r (the radial coordinate) and theta (the angular coordinate, often called the polar angle) are defined in terms of Cartesian coordinates.
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Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height (z) axis.
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HORIZONTAL CURVE FORMULAS ; L · = Length of Curve, from P.C. to P.T. ; T · = Tangent Distance ; E · = External Distance ; R · = Radius ; L.C., = Length of Long Chord.
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Sep 28, 2022 · Application: When dimensioning a circular feature using its radius, the radius symbol, “R,” is placed before the size dimension as shown below.