Concave Polygon
A concave polygon has at least one interior angle greater than 180 degrees, creating an inward-pointing "dent."
Definition
A concave (or non-convex) polygon has at least one corner that points inward, creating a "dent," with at least one interior angle greater than $180^\circ$ (a reflex angle). At least one diagonal lies partially or entirely outside the polygon, and while the exterior angle sum is still $360^\circ$ for simple polygons, some exterior angles are negative at the reflex vertices. Formally, a simple polygon is concave if and only if it is not the intersection of half-planes; the dual notion to the convex hull is the concave hull, or $\alpha$-shape, which captures the "true" boundary of a point cloud including its concavities.
Example
An arrow shape, a star, and an L-shape are all concave polygons; tracing the inside corner of an L-shaped room reveals a reflex angle that makes the shape concave. An L-shaped polygon has $6$ vertices with interior angle sum $(6-2)\times180=720^\circ$: five angles of $90^\circ$ and one of $270^\circ$ sum to $5\times90+270=720$, with the reflex $270^\circ=360-90$ confirming the "missing" right angle is internalized as a reflex.
Key Insight
If a rubber band stretched around the shape does not touch every vertex, skipping over some because they dent inward, the polygon is concave, the opposite of convex. Triangulation is more complex for concave polygons since diagonals from one vertex may fall outside, requiring special ear-clipping algorithms used in 3D modeling and game physics; alpha-shapes and concave hulls play a similar role in reconstructing the true boundary of geographic data like coastlines and building footprints, with applications in robotics and medical imaging.