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Polygons - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Polygons are closed plane figures formed by three or more line segments. If a figure is open or curved, it cannot be considered a polygon. Concave polygons have at least one diagonal that does not pass through the interior of the polygon; all of the diagonals in a convex polygon are contained within the figure. Equiangular polygons have all angles congruent; equilateral polygons have all sides congruent.

Polygons are discussed throughout Geometry, so it's important to know their characteristics. First of all, a polygon is a closed figure. So if I drew a figure down here, this would not be a polygon because as you can see, there's this open space here. However, if I drew in a line segment then it would be a polygon because now it is closed. The second thing about polygons is that it has at least three sides. If you have less than three, you can't close any figure. And none of the sides are curved. So if we go back to this figure I drew originally, and let's say instead of drawing a straight line, I drew some sort of curved line. This would not be a polygon because, yes three are the lines are straight, but you have one curved side, so this would not be a polygon.
And the third thing is that they're classified by the number of sides. So what you're going to need to do is memorize a table, that's probably found in your textbooks, that lists the number of sides and the name for that polygon. This is something that you're going to have to memorize because you're going to see it on tests and quizzes throughout geometry.
Now there are two different types of polygons, convex and concave. And the difference is in the diagonals. Remember a diagonal is a segment that connects non consecutive vertices. So in this polygon, if I drew in my diagonals, you'll notice that all of those diagonals are contained within that polygon. So it would be considered convex.
To be concave, you need at least one diagonal that is outside of your polygon. So here, I can draw in a diagonal that is not within this polygon. It just so happens that we have two in this polygon. But you only need one to be considered concave.
Now the last key thing about polygons is how do you name them. Well that's pretty easy because all you have to do is pick one vertex. So let's say I pick D. If I start with D, I can go in, clockwise or counter clockwise, but going to have to be consecutive. So if I start with D, there are two ways that I can name this pentagon. I can go D, C, then B, then A, and then E. So that's one way to name that polygon.
The other one, if I'm starting with D, is to go in the opposite direction. So I could say this is D, E, A, B, and C. So when you're naming a polygon, pick any vertex and go in a consecutive order, either counter clockwise or clockwise. Remember when you're asked that on your true and false questions, the key things about polygons are closed figure, three sides, none of which are curved, and you classify it by how many sides it has.

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