Halfchord

Halfchord is a term used in geometry and trigonometry to denote the half-length of a chord of a circle subtending a given central angle. If a circle of radius R has a chord that subtends a central angle θ, the full chord length is c = 2R sin(θ/2). The half-chord length is h = c/2 = R sin(θ/2). On the unit circle (R = 1), the half-chord equals sin(θ/2) and can also be written as sqrt((1 − cos θ)/2) via the half-angle identity.

It is important to distinguish the half-chord from the perpendicular distance from the center to the chord. That distance, d, equals R cos(θ/2). Thus, while h relates to the half-angle, d relates to the complementary half-angle.

Historically, half-chord, sometimes called semi-chord, appeared in trigonometric tables and geometric discussions before sine tables were standard. The concept remains a straightforward geometric interpretation of sin(θ/2) and appears in problems involving chords and inscribed angles, as well as in trigonometric substitutions where half-angles are natural.

Applications of the half-chord concept appear in circle geometry, surveying, and navigation problems that involve chord lengths and inscribed angles. It provides a simple link between central angles and chord measurements, particularly in unit-circle calculations.

See also: chord; half-angle; chord length formula; trigonometric half-angle identities.