subtending

Subtending is a geometric term describing how one object, such as a line segment, chord, or arc, relates to a point or another object by forming an angle or serving as the boundary that defines that angle.

In plane geometry, a segment AB subtends an angle at a point P if the lines PA and PB form the angle ∠APB. A chord AB of a circle subtends a central angle ∠AOB at the circle’s center O, and it also subtends inscribed angles ∠APB at any point P on the circle. The Inscribed Angle Theorem states that any inscribed angle subtending the same arc AB is half the measure of the corresponding central angle subtending that arc.

The locus of points P from which AB is seen under a fixed angle θ (0 < θ < 180°) is an arc of a circle with AB as a chord; there are two symmetric arcs on opposite sides of AB. A notable special case is Thales’ theorem: if AB is a diameter, any point P on the circle with AB as diameter sees AB under a right angle, so ∠APB = 90°.

Beyond circles, subtending is often used to describe the apparent angular size of an object as seen from a given location: a nearby object subtends a larger angle than a distant one. The term can also be used in broader geometric contexts to describe how a line or region relates to a point or other figure via the angle it forms.

See also: central angle, inscribed angle, chord, circle, arc, Thales’ theorem.