Reflex Angle
GeometryA reflex angle measures greater than 180 degrees and less than 360 degrees.
Formula
180 < \text{angle} < 360^\circ
Definition
A reflex angle is greater than $180^\circ$ but less than a full $360^\circ$ circle. It is the "big" angle on the outside when you draw two rays from a point.
Example
If you cut a small slice of pizza, the remaining large portion of the pizza forms a reflex angle at the center. A $270^\circ$ angle is reflex. At 7 o'clock, the large angle swept from the 12 to the 7 going the long way is a reflex angle.
Key Insight
Every angle between two rays actually creates TWO angles: one less than $180^\circ$ (the regular one) and one greater than $180^\circ$ (the reflex one). They always add up to $360^\circ$.
Definition
A reflex angle has measure strictly between $180^\circ$ and $360^\circ$. For any non-straight angle, its reflex counterpart equals $360^\circ$ minus the original angle. Reflex angles appear in concave polygons (where at least one interior angle is reflex) and in circle geometry when discussing major arcs.
Example
If an angle measures $70^\circ$, its reflex angle is $360 - 70 = 290^\circ$. A concave polygon has at least one interior reflex angle. The reflex angle of a right angle is $360 - 90 = 270^\circ$.
Key Insight
In circle geometry, the central angle for a major arc is a reflex angle. The inscribed angle theorem still holds: an inscribed angle is half its intercepted arc, even when the arc is major (greater than $180^\circ$).
Definition
A reflex angle $\theta$ satisfies $\pi < \theta < 2\pi$ radians. In directed angle conventions (mod $2\pi$), every pair of rays defines a directed angle in $[0, 2\pi)$. In the context of concave polygons, a reflex interior angle means the polygon's interior lies "outside" the usual turn at that vertex.
Example
The exterior angles of a concave polygon can be negative (when the interior angle is reflex). The sum of exterior angles of any simple polygon is still $360^\circ$, but some exterior angles are negative for concave polygons.
Key Insight
Reflex angles highlight the importance of orientation in geometry. When computing areas using the shoelace formula or defining winding numbers, the distinction between a reflex angle and its non-reflex counterpart determines the sign of a contribution, connecting angle measure to signed area.