Overview. In geometry, the radius of a circle or sphere is the straight-line distance from the center to the boundary. For a circle it reaches the perimeter; for a sphere it reaches the surface. The radius is a fundamental linear measure that determines size and scale for round shapes. Visually it is often shown as a line segment inside the figure from the center point to the edge.

Basic relations and common formulas

The radius is half the diameter: if d denotes the diameter and r the radius, then r = d/2 and d = 2r. The radius sets other basic quantities for circles and spheres. For a circle with radius r, the circumference (the length around the circle) equals 2πr and the enclosed area equals πr². For a sphere of radius r, the surface area equals 4πr² and the volume equals (4/3)πr³. These relations make the radius the primary parameter when deriving sizes, scaling diagrams, or computing properties of circular objects.

Properties and useful variations

  • Circumradius and inradius: In polygons, the circumradius is the radius of the circumscribed circle that passes through all vertices, while the inradius is the radius of the inscribed circle tangent to all sides.
  • Radius of curvature: In differential geometry and engineering, radius of curvature describes how sharply a curve bends at a point; smaller radii correspond to tighter bends.
  • Higher dimensions: The same concept extends to hyperspheres in higher-dimensional spaces; many formulas include the radius as the scaling factor for area and volume analogues.

History and terminology

The word "radius" comes from Latin, originally meaning a spoke of a wheel or a ray. Classical geometers used the term for line segments drawn from centre points to circles. Over centuries the radius became a standard descriptor in Euclidean geometry and later in analytic and differential geometry where it remains central to the study of circular and spherical symmetry. Historical treatments often introduced radius while explaining diameters, chords, tangents, and the notion of a center.

Applications and examples

Practical uses of the radius appear across science and everyday life. Engineers use radii to design wheels, gears, pipes and arches; surveyors use radii to set out circular boundaries; astronomers measure orbital radii; and in computer graphics the radius defines hit zones, collision spheres, and rounded corners. For example, a circle with radius 5 units has diameter 10 units, circumference 10π units, and area 25π square units. Many technical formulas substitute r directly to simplify derivations or to express scaling laws.

It is helpful to distinguish radius from related segments: a diameter is a chord passing through the center whose length is twice the radius; a chord is any segment joining two points on the boundary and equals 2r·sin(θ/2) when subtending an angle θ at the center. In coordinate geometry, a circle with center (h,k) and radius r satisfies (x − h)² + (y − k)² = r², which makes the radius explicit in algebraic descriptions. These distinctions show how the simple notion of radius interacts with metrics, angles and algebraic representations.

Further reading and references. For introductory treatments see standard texts on plane geometry and elementary calculus that explain circle and sphere properties in more detail. Related entries discuss diameter, circumference, area, and volume; you can also follow links on circle, sphere, perimeter, surface, diameter, circumference and area.