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Radius is a fundamental concept in geometry, referring to the distance from the center of a circle or sphere to any point on its circumference or surface. It is typically denoted by the lowercase letter 'r'. The radius is half the length of the diameter, which is the distance across the circle or sphere passing through its center.

The radius is crucial for defining and calculating various properties of circles and spheres. For instance, the area of a circle is calculated using the formula A = πr², where π (pi) is a mathematical constant approximately equal to 3.14159. Similarly, the circumference, or the distance around a circle, is given by C = 2πr.

In three-dimensional geometry, the radius of a sphere determines its volume (V = (4/3)πr³) and surface area (SA = 4πr²). The concept of radius also extends to other geometric shapes, such as the radius of curvature for curves and surfaces, which describes how sharply a curve bends at a particular point.

Beyond geometry, the term "radius" is used metaphorically in various contexts to denote the extent or reach of something, such as the radius of influence or the radius of operation. However, in its primary mathematical sense, it remains a precise measure of distance from a central point.