General · EdgepediaSeptember 17, 20266 min read

Radius

A radius is the straight line from the center of a circle to any point on its circumference, or the length of that line; in modern English the word covers both the segment and its measure¹. The plural is radii, and the standard symbol is r (or R when several radii appear in one argument). The radius is one of the two fundamental dimensions of a circle, the other being the diameter, which is twice the radius⁴. The same idea extends to spheres, where the radius joins the center to any point on the surface, and to higher-dimensional analogs such as balls and disks.

Key fact	Detail
Definition	The distance from the center of a circle to any point on the circle; all radii of one circle are equal⁵
Relation to diameter	The radius is half the diameter; a diameter passes through the center and has both endpoints on the circle⁴
Circumference	$C = 2\pi r$ , equivalently $C = \pi d$ ⁴
Area and volume	Circle area $A = \pi r^{2}$ ; sphere surface $S = 4\pi R^{2}$ and ball volume $V = \frac{4}{3}\pi R^{3}$ ³
Coordinate equation	$(x-h)^{2} + (y-k)^{2} = r^{2}$ for a circle of radius r centered at $(h,k)$ ⁵
Curvature	A circle of radius a has constant curvature $\kappa = 1/a$ ; the radius of curvature of a general curve is $\rho = 1/\kappa$ ⁶
Word history	Latin radius meant a staff, rod or stake, and by extension a ray of light; first used for the circle line by Peter Ramus in 1569, and in English mathematical writing by Hobbes in 1656¹

Definition and basic relations

A circle is the set of all points in a plane at a fixed distance from a given point; the given point is the center and the fixed distance is the radius⁵. Because every point of the circle satisfies the same distance condition, all radii of one circle are equal, and so all diameters are equal as well².

The radius enters the standard measurement formulas. The circumference is $C = 2\pi r$ , where $\pi \approx 3.141592654$ is the constant ratio of circumference to diameter; because that ratio is constant, the formula holds for a circle of any size⁴. The area is $A = \pi r^{2}$ , which can be motivated by cutting a circle into sectors and rearranging them into an approximately rectangular shape with sides $\pi r$ and r⁸. For a sphere of radius R, the surface area is $S = 4\pi R^{2}$ and the volume of the enclosed ball is $V = \frac{4}{3}\pi R^{3}$ ; Archimedes derived these results in On the Sphere and Cylinder (ca. 225 BC) and also established that the sphere's volume is two thirds that of the circumscribed cylinder³.

In coordinates. Placing the center at $(h,k)$ , the distance formula turns the definition into the standard equation of a circle,

$(x-h)^{2} + (y-k)^{2} = r^{2}.$

When the center is the origin this reduces to $x^{2} + y^{2} = r^{2}$ ⁵. The three-dimensional analog is the sphere $x^{2} + y^{2} + z^{2} = R^{2}$ centered at the origin, or $(x-x_{0})^{2} + (y-y_{0})^{2} + (z-z_{0})^{2} = R^{2}$ centered at $(x_{0}, y_{0}, z_{0})$ ³.

History of the term

The Greeks had no special word for the radius in either the segment or the length sense, in contrast with the diameter, for which they did. Euclid's Elements defines a circle through the distance from the center but never names the radius; the phrase used is "that from the center," and in Euclid's postulate 3 the term for "distance" was considered sufficient¹^•². Archimedes called the radius "ek tou kentrou," meaning the line from the center¹.

Latin and English adoption. The Latin word radius originally meant a staff, rod, or stake, and by extension a ray or beam of light¹. Before "radius" was adopted for the circle line, writers used "semidiameter": the Latin term appears in the Ars Geometriae of Boethius around 510, and the English word appears in 1551 in Robert Recorde's Pathway to Knowledge, where he defines the semidiameter as the half of the diameter measured from the center to the circumference¹. The French humanist Peter Ramus (1515 to 1572) used radius for the circle line in his 1569 Scholarium mathematicarum, writing "Radius est recta a centro ad perimetrum"¹. The Oxford English Dictionary's earliest reference to radius as a mathematical term in English is Thomas Hobbes in 1656, asking whether "the radius that describes the inner circles" equals "the radius that describes the exterior"¹.

The word then generated a family of technical terms. The radius of convergence of a power series appears in English in 1891, in George Lambert Cathcart's translation of Axel Harnack's German calculus textbook, and the radius of curvature was treated by Euler in his 1748 Introductio in analysin infinitorum, with earlier use possible by Christiaan Huygens and Isaac Newton¹.

Radius of curvature

In differential geometry the radius generalizes from circles to arbitrary curves. At a point on a smooth curve, the circle that best approximates the curve is the circle of curvature, also called the osculating circle, a name due to Leibniz, from the Latin "osculare," to kiss. The radius of that circle is the radius of curvature $\rho$ , and the curvature is its reciprocal, $\kappa = 1/\rho$ ⁶. A circle of radius a is its own osculating circle everywhere and has constant curvature $1/a$ ⁶. Large curvature means a small radius of curvature and a sharply turning curve; small curvature means a large radius and a nearly straight curve, with straight lines having curvature exactly zero⁶.

Measuring a radius: the Earth example

The radius of a large sphere can be found from small-scale measurements by combining shadow geometry with a surface distance. A classroom demonstration at the University of Maryland models the sunlit Earth with a disc of radius R carrying two radial tabs illuminated by a parallel beam of light. Measuring the tab length b, its shadow length a, and the surface distance S between the tabs gives, by similar triangles, $S/R = a/b$ , so

$R = \frac{bS}{a},$

which can be checked against the directly measured radius of the disc⁷. The same trigonometric scheme underlies the classical determination of the Earth's radius from shadow angles at different latitudes.

Common errors in using the radius

Mathematics education research documents recurring mistakes involving the radius. Students sometimes treat $r^{2}$ in the area formula as $r \times 2$ , use the diameter in place of the radius in $A = \pi r^{2}$ , or square the diameter instead of the radius⁹. Confusing radius and diameter is listed among the most frequent circle-related errors, and teachers are advised to have students derive the formulas rather than only substitute into them⁹.

References

Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Geometry and topology › Elementary and Euclidean geometry › Plane figures

Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026

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