diameters

Diameters are distances across a circle, sphere, or, in more general terms, the greatest distance between any two points in a set. In everyday geometry, the diameter is most commonly discussed in circles and spheres, where it has a precise, simple relationship to the radius.

In a circle, a diameter is a line segment whose endpoints lie on the circle and which passes through the center. There are infinitely many diameters; every diameter has the same length, equal to twice the circle’s radius. If r denotes the radius and d denotes the diameter, then d = 2r. The circle’s area is A = πr^2 = (π/4) d^2, and its circumference is C = 2πr = πd. Thales’ theorem states that any angle subtended by a diameter is a right angle.

In a sphere, the diameter is the greatest distance between any two points on the surface, and it equals twice the sphere’s radius, D = 2R. Formulas connecting diameter to other quantities include the volume V = (π/6) D^3 and the surface area S = π D^2.

Beyond circles and spheres, the term diameter can refer to the diameter of a set in a metric space: the supremum of the distances between all pairs of points in the set. For finite sets, this is simply the maximum distance observed.

In practical use, diameter is a standard measure for circular objects and features such as coins, pipes, lenses, and architectural elements, typically expressed in the same units as other linear dimensions.

See also: Radius, Circumference, Thales’ theorem.