What Is a Concave Polygon?

How the inward dent changes the figure

Concavity matters because it changes how diagonals and slicing lines behave. Some diagonals fall outside the polygon, and a line through the figure can intersect the boundary more than twice.

This topic is important because many polygon rules are first learned on convex figures. Seeing the concave case clearly helps students understand which ideas still work and which need extra care.

A concave polygon can still have a perfectly ordinary number of sides and can still have a well-defined interior-angle sum. The difference is the inward turn. When checking a drawing, trace the boundary in order and pause at each vertex; the reflex corner will reveal the concavity more reliably than the overall outline.

The diagonal from one side of an inward dent to another may pass outside the polygon. That is a useful visual contrast with convex figures, where every diagonal remains inside. The difference comes from the single reflex angle, not from the number of sides.

• A concave polygon has at least one interior angle greater than 180 degrees.
• A concave polygon has at least one reflex interior angle.
• Some diagonals of a concave polygon lie outside the polygon.
• No triangle can be concave.

When concavity is important

• Use concavity when classifying polygons whose boundaries fold inward.
• Use it in diagonal and triangulation work where outside segments matter.
• Use it to understand why some polygon diagrams require more care than convex ones.

Concave-shape checks

• Do not call a polygon concave just because it looks unusual; confirm that an interior angle exceeds one hundred eighty degrees.
• Do not assume all irregular polygons are concave.
• Do not apply inside-diagonal assumptions from convex polygons without checking the shape first.