Concave Polygon
GeometryA concave polygon has at least one interior angle greater than 180 degrees, creating an inward-pointing "dent."
Definition
A concave polygon has at least one corner that points inward, creating a "dent" in the shape. At least one interior angle is greater than $180^\circ$ (a reflex angle).
Example
An arrow shape, a star, and an L-shape are all concave polygons. If you trace the outside of an L-shaped room, the inside corner creates a reflex angle and makes the shape concave.
Key Insight
If a rubber band stretched around the shape does NOT touch every vertex (it skips over some because they dent inward), the polygon is concave. Concave is the opposite of convex.
Definition
A concave (or non-convex) polygon has at least one interior angle greater than $180^\circ$ (a reflex angle). At least one diagonal lies partially or entirely outside the polygon. The exterior angle sum is still $360^\circ$ for simple polygons, but some exterior angles are negative (turning backward at reflex vertices).
Example
An arrowhead polygon: most angles are acute or obtuse, but one interior angle is $270^\circ$ (reflex). A diagonal drawn to include this vertex will pass outside the polygon. The angle sum theorem still gives $(n-2) \times 180$ for all interior angles, even if some are reflex.
Key Insight
Triangulation is more complex for concave polygons than convex ones: you cannot simply draw diagonals from one vertex because some diagonals may fall outside. Special ear-clipping algorithms handle concave triangulation. This complexity matters in 3D modeling and game engine physics.
Definition
A simple polygon is concave iff it is not convex: i.e., it is not the intersection of half-planes, some interior angles exceed $\pi$, and some diagonals lie outside. The dual notion to the convex hull is the concave hull ($\alpha$-shape): for a parameter $\alpha$, the $\alpha$-shape captures the "true" boundary of a point cloud including concavities. A star polygon is a non-simple generalization.
Example
An L-shaped polygon has $6$ vertices, interior angle sum $= (6-2) \times 180 = 720^\circ$. Five angles are $90^\circ$ and one is $270^\circ$: $5\times90 + 270 = 720$. The reflex angle of $270^\circ = 360 - 90$, confirming the "missing" right angle is internalized as a reflex.
Key Insight
Concave polygons arise naturally in geographic data (coastlines, building footprints) where purely convex shapes miss important features. Alpha-shapes and concave hulls are used in computational geometry to reconstruct the "true" boundary of a point set - applications include robotics, medical imaging, and geographic information systems.