Concave and Convex Quadrilaterals at a Glance

Gee willikers! What happened to Pacman?

He looks so rigid and angular, nothing like his usual well-rounded self. Could it be that this isn't Pacman, but his evil twin—Quadman? Well, it only makes sense. After all, he's a quadrilateral, and look! Even his food dots are quadrilaterals. Clearly, we're in some major quadrilateral territory.

There's an important difference between Quadman and his food. (We aren't talking about the obvious polygonal predator-versus-prey situation going on here.) This Quadman is a perfect example of a concave quadrilateral, while his food is made of convex quadrilaterals.

Concave quadrilaterals are those that have a cavity, or a cave. In the game, Quadman's mouth is the "cavity" we're talking about. Convex quadrilaterals, like the more familiar squares and rectangles, don't have a cavity. Of course, as mathematicians, we might want to come up with a better definition than "has a cavity" or "doesn't have a cavity." It sounds too much like a painful trip to the dentist.

Let's take a closer look at Quadman and his food.

When we draw the diagonals on Quadman, we see that one of his diagonals (the horizontal one) lies inside his body, but the other one doesn't. If we do the same for his food, we see that both diagonals are completely contained within the shape itself. That's the difference between concave and convex.

So let's define them again, take two. A convex quadrilateral has both diagonals completely contained within the figure, while a concave one has at least one diagonal that lies partly or entirely outside of the figure. Those are good definitions, but the concave shapes having a cavity or cave is probably an easier way to remember it.