Lesson and Practice: Convex, Concave and Self-Intersecting Polygons

Simple and self-intersecting polygons

In the app below you can see a polygon: polygon is a word derived from Greek, and means "many angles". A polygon can be simple, when its boundary does not cross itself, otherwise it's self-intersecting. Drag the vertices of the polygon (green points) and create simple and self-intersecting figures. The intersection points will be shown in orange.

Convex and Concave Polygons

When polygons are simple, we can characterize them further, as convex or concave. Explore the applet below, create a few different convex and concave polygons, observe the measure of the angles and the position of the diagonals with respect to the portion of the plane enclosed within its sides. Formulate your own conjecture about how to differentiate a convex and a concave polygon, depending on these properties of the figure.

Convex polygons and interior angles

What can you say about the measures of the interior angles of a convex polygon? Do they all have a common characteristic?

Concave polygons and interior angles

What can you say about the measures of the interior angles of a concave polygon? Do some of them have a common characteristic?

Convex polygons, concave polygons and their diagonals

Describe the difference between the position of the diagonals of a convex and of a concave polygon, with respect to the portion of the plane enclosed within its sides.