Reflex angle

Characteristics and measurement

Reflex angles contrast with acute, obtuse and straight angles by exceeding a straight angle (180°) but not completing a full turn (360°). If the smaller of the two angles formed by two rays has measure θ where 0° < θ < 180°, the corresponding reflex angle can be written as 360° − θ. In radians the same relation is 2π − θ (with θ measured in radians).

Common examples and interpretation

Typical numerical examples include 200°, 270° and 359°. A 270° angle is often visualized as three quarters of a full rotation. In contexts that distinguish direction, an oriented angle may be given by a signed measure: the reflex angle in the positive (counterclockwise) sense is simply the larger positive measure between the two rays.

Uses and occurrences

• Geometry and polygons: reflex interior angles occur in non-convex (concave) polygons; an interior angle greater than 180° indicates a reflex vertex.
• Graphics and animation: rotations often use angles greater than 180° to indicate long turns or sweeps.
• Navigation and robotics: planning a path sometimes requires turning through a reflex angle, especially when following a specified orientation rather than the shorter rotation.

Construction and related facts

Standard Euclidean tools can construct a reflex angle by first drawing the smaller angle and then subtracting it from a full circle, or by rotating a ray past the straight position. Reflex angles can be bisected with the same classical procedures used for other angles, though practical construction often involves drawing auxiliary lines or using protractors marked for full 360° measurement.

Notable distinctions: reflex angles are not classified as convex; they identify concavity at a polygon vertex and are handled differently in angle-sum calculations where exterior or interior angles are considered. When working with oriented angles or modular arithmetic on angles, it is common to reduce measures modulo 360° or 2π, which can convert a reflex measure to an equivalent smaller negative or positive value depending on the chosen principal interval.