Welcome to introduction to polygons.
The goals of this video are to define a polygon and then classify different
types of polygons.
A polygon is a plane figure made up of segments
joined at the endpoints. Here we see four examples of polygons.
I do want to point out that there is some disagreement about
this last polygon. Some definitions state that the segments cannot intersect each
other,
but we'll address this issue in the next slide. The points where the sides join are called
the vertices.
Looking at this triangle here, two sides join at this point here,
this would be a vertex. Vertex is singular for
vertices. Here's a vertex and here's a vertex.
And the segments are called the sides, or edges of the polygon.
So this triangle here has three edges,
or three sides, one here,
one here, and one here. Here are a couple of examples
of figures that are not polygons. This is not a polygon because
it is not formed by line segments. This
edge here is a curve. And this is not a polygon the because the four edges are
not joined together
to form a bounded region. So sticking to the previous definition of a polygon there are
two types a polygons:
a simple polygon has one bound region, where the sides do not intersect.
And a complex polygon has more than one
bounded region, because the sides do intersect.
So these two are examples of simple polygons,
and here's an example of a complex polygon.
And this complex polygon has three
bounded regions. One here, one here,
and one here, for a total of three bounded regions.
Now let's talk about the diagonals of a polygon.
The diagonals of a polygon are the line segments that connect
non adjacent vertices of a polygon.
So looking at this polygon here, if we select one vertex,
the diagonals from this vertex here, would be all of the segments connecting
this vertex with any nonadjacent vertex. Well adjacent vertices
are the vertices that are next to this vertex. So this would be an adjacent
vertex,
and so would this one. So the diagonals from this point would be
this segment here,
and this segment here. If I wanted to sketch diagonals from this vertex, I would
do the same.
This is not a diagonal, this would be a side. So I connect
this vertex with this nonadjacent vertex,
and this vertex with this nonadjacent vertex.
Let's go and see if we can sketch the rest of them.
When we sketch the diagonals from this vertex here, we have to be a little bit careful because
we will start to experience some repetition. Notice this diagonal is already
sketched,
so now we'll connect this vertex to this vertex,
and then this vertex to this vertex. And now we'll go to this vertex here.
This diagonal is already sketched, this diagonal is already sketched.
We still have to sketch this diagonal here.
According to this vertex here, notice all the diagonals are already drawn.
So from each vertex
we can see there are three diagonals. Now for this polygon, let's just go ahead and sketch
the diagonals from one of the vertices.
If we were to select this vertex here,
this vertex and this vertex would be the adjacent vertices.
So the diagonals would be the segments to the remaining vertices,
here,
here, and so on.
Some of these are hard to see because they do overlap. But what I want to point out here
is that notice that
some of the diagonals pass outside the polygon,
whereas in our first example, all the diagonals were inside the polygon.
So these two polygons represent two different types
of simple polygons. Simple polygons are either convex,
or concave. The diagonals of a convex polygon are always
inside the polygon. However, at least one diagonal
of a concave polygon will pass outside the polygon.
So this would be a simple convex
polygon because all the diagonals were inside the polygon,
as we saw in the previous screen. And this would be a simple concave polygon
because we saw that some of the diagonals
would pass outside the polygon. You can also think of a concave polygon
as two sides caving toward the inside of the polygon.
So these two short sides here, cave toward the inside of the polygon,
and therefore this is a concave polygon. The last topic of this video will be
regular polygons. A regular polygon is a polygon
where the sides are equal length, or equilateral and angles of equal measure,
or equal angular. And here we see four examples
of regular polygons. Remember if all the sides are the same length, we can put
the same number of hash marks on each side,
to indicate the sides have the same length.
And we can also put a single arc
through each angle to identify all of the angles are equal measure.
And we could do the same on the remaining three polygons. So if I put two
hash marks here,
and it's a regular polygon, I'd have to put two hash marks
on each side indicating all of the sides are the same length.
When it comes to the angles, I can either put
the same number of arcs, or make a single arc,
and then put the same number of hash marks in each arc.
So if I put two hash marks through this arc, I'd have to put two hash marks through
each of the other arcs. And we'll go and stop here for this intro video.
I hope you found this helpful.