Subtend

Subtend is a term used primarily in geometry to describe how a geometric object relates to an angle or a point. When a line segment, chord, or arc lies in a circle, it may subtend an angle at a given point, or subtend an arc or a central angle at the circle’s center. The phrase is often used with the words “angle” or “arc,” as in “the chord subtends an angle” or “the chord subtends the arc.”

In a circle, a chord AB subtends two related things: a central angle AOB at the circle’s center O, and an inscribed angle ACB at any point C on the circle’s circumference on the opposite side of AB. The inscribed angle theorem states that the inscribed angle is half the measure of the central angle that subtends the same chord. The arc AB is the portion of the circle between A and B, and the chord AB subtends that arc, while the arc subtends the central angle.

Useful formulas arise from these relationships. If the circle has radius R and the central angle subtending chord AB is θ (in radians), then the chord length AB = 2R sin(θ/2), and the length of arc AB = Rθ. Consequently, the inscribed angle ∠ACB subtending AB equals θ/2.

Subtend is also used more generally to describe how a boundary or line segment can lie opposite or define an angle or region from a specified viewpoint. The term comes from the Latin subtendere, meaning to stretch underneath. Related concepts include inscribed angle, central angle, chord, and arc.