The radius is the distance from the center of a circle (or sphere) to any point on its circumference (or surface).
The radius is a fundamental measurement of a circle. It is half the diameter: r = d/2. The formula connecting radius to circumference is C = 2πr, and to area is A = πr². In a sphere, the radius determines surface area (4πr²) and volume (4πr³/3). Radii are used in coordinate geometry, trigonometric functions on the unit circle, and countless engineering applications.
The word "radius" is Latin for "ray" or "spoke of a wheel." Euclid defined the radius in his "Elements." The unit circle (radius = 1) became fundamental in trigonometry, developed by Hipparchus and Ptolemy.