a convex versus a concave polygon

Picture stretching a rubber band around a shape. If the band hugs the figure everywhere — touching every corner with no dents — the polygon is convex, like a triangle, a square, or a regular hexagon. If the band springs across a notch and leaves a gap, the figure has a 'cave' pushed inward; it is concave (also called non-convex), like a five-pointed star's outline or an arrowhead.

Two equivalent precise tests pin this down. First test: a polygon is convex when, for every pair of points inside it, the whole straight segment joining them stays inside; in a concave polygon you can find two interior points whose connecting segment pokes outside the figure. Second test, on the angles: a polygon is convex exactly when every interior angle is less than 180 degrees. A concave polygon has at least one reflex interior angle — an angle greater than 180 degrees — and that reflex corner is the 'dent'. A third handy test: extend any side of a convex polygon to a full line, and the entire polygon lies on one side of that line; for a concave polygon some such line cuts through the interior.

Convexity matters far beyond naming. Many clean theorems — the interior-angle-sum formula (n-2)·180, the exterior-angle-sum of 360 degrees, a polygon's having a well-behaved inscribed or circumscribed circle — are stated most simply for convex polygons, and several break or need care for concave ones. The caveat to remember: 'concave' does not mean 'irregular' and 'convex' does not mean 'regular'. An L-shaped tile is concave but perfectly regular-looking in its right angles; a slanted, lopsided quadrilateral can be convex yet have no equal sides at all.