radius2

Radius2 refers to the squared radius, commonly written as r^2, where r is the radius of a circle, sphere, or higher-dimensional sphere (hypersphere). The term is used to describe size without taking the square root, and it appears frequently in both theoretical and applied contexts.

In geometry, radius2 appears in the standard equations of circles and spheres. For a circle with center (h, k), the equation is (x − h)^2 + (y − k)^2 = r^2. The squared form is convenient because it avoids square roots when measuring distances. In three dimensions, a sphere with center (a, b, c) satisfies (x − a)^2 + (y − b)^2 + (z − c)^2 = r^2. The value r^2 also appears in formulas for surface area and volume: the surface area of a sphere is 4πr^2, and the volume is (4/3)πr^3, where r^2 is a component of the expressions.

In computational contexts, radius2 is often stored instead of r to speed up calculations. For example, to test whether a point is inside a circle, one can compare dx^2 + dy^2 to r^2, avoiding the need to compute a square root. This approach extends to higher-dimensional distance tests and collision detection in computer graphics and physics simulations.

Beyond pure geometry, r^2 features in physics and statistics. In Newtonian gravity and electromagnetism, force scales with 1/r^2. In statistics and machine learning, squared radii can arise in kernels, distance metrics, and normalization schemes. Radius2 is thus a fundamental, frequently used quantity across disciplines.