Subtended

Subtended describes a geometric relationship in which one object lies under the lines, angle, or arc defined by another. In common mathematical usage, an object is said to subtend an angle, arc, or chord, meaning the angle or arc is intercepted by the endpoints or boundaries of that object.

In circle geometry, a line segment with endpoints on a circle subtends a central angle at the circle’s center and also subtends an arc of the circle between those endpoints. The inscribed angle that also subtends the same arc lies on the circle and equals half the measure of the central angle. If the circle has radius R and the central angle is theta (in radians), the chord length is c = 2R sin(theta/2) and the subtended arc length is s = R theta.

The phrase also applies to angular size: an object at a given distance subtends a certain angle at the observer. For a straight object of length L at distance d, the exact angular size is theta = 2 arctan(L/(2d)); for small angles, theta ≈ L/d when theta is measured in radians.

In three dimensions, a surface or patch can subtend a solid angle at a point, measured in steradians, reflecting how large the object appears to the observer.

Etymology traces to Latin subtendere, meaning to stretch up from underneath or to place beneath. The term is used across geometry, trigonometry, and optics to describe how geometric figures determine or are determined by angles, arcs, and viewing directions.