Welcome to a lesson on classifying polygons.
The goals of the video are to classify a simple polygon based upon a number of
sides,
and then also to investigate the number of diagonals a polygon has.
For a quick review, in the intro video we defined a polygon
as a plane figure made up of segments joined at the endpoints.
And we said there were two types of polygons:
simple polygons and complex polygons.
Simple polygons have sides that do not intersect each other,
as we see here. And a complex polygon
has sides that intersect, as we see here in this polygon.
And then there are two classifications of simple polygons:
convex polygons and concave polygons.
These two were convex polygons,
because all the diagonals would be inside
the polygon. And this would be a concave polygon,
because some of the diagonals would be outside the polygon.
but you can also think of it as two or more sides
caving in to the interior of the polygon, as we see here in this region.
And then these two figures down here we're not polygons.
This one because one edge is curved, not a line segment,
and this one because not all the segments are joined at the endpoints.
Now there are specific names for polygons
based upon the number of sides the polygon contains.
The polygon has three sides
it's called a triangle, four sides a quadrilateral,
five sides a pentagon, six sides
a hexagon, seven sides a heptagon,
eight sides an octagon, nine sides a nonagon,
ten sides a decagon. And there are more names,
but just remember all of these are special names for the bigger group
of polygons. So they're all polygons but within the set of polygons,
we can name each polygon by the number of sides it contains.
And we also talked about diagonals
in the introductory vide. But let's do a quick review before we take a look at
determining
how many diagonals a polygon actually has. The diagonals of a polygon are the line
segments that connect
nonadjacent vertices of a polygon. So the vertices of the polygon
are where the segments join each other, this has five
vertices. And the segments making up the polygon are called the edges,
or sides. So if we want to discuss the diagonals from this vertex here,
these two would be the adjacent vertices,
so we'd have a diagonal here and a diagonal here from this vertex.
Now if we go around to each to each vertex we can sketch additional diagonals
but we'll go ahead and stop here. But notice for our pentagon,
there will be two diagonals from each vertex.
Let's go ahead and sketch the diagonals from one vertex of this concave hexagon.
If we select this vertex here,
we would have a diagonal here, here,
and here. Notice that set for a hexagon, there are three diagonals from each vertex.
So let's go ahead and take a look at the diagonals from a variety of polygons.
Lets see if we can compare the number of sides to the number of diagonals.
Well here we have a triangle with three sides, but if we select a vertex, the
remaining vertices are adjacent
to the selected vertex, so if there are three sides
there are actually zero diagonals. For a square,
if we select this vertex, there's only one nonadjacent vertex and that would be
this one here. So there's one diagonal from this vertex,
there will be another diagonal from this vertex here.
There are no other diagonals for this
quadrilateral, so if there are four sides
there are two diagonals. And we started this one on the previous screen,
we said from each vertex there will be two diagonals. So there's
one here, and one here from this vertex. One here and one here from this
vertex,
and then when we go to this vertex there's already one sketched here, so we do have to sketch one more
connecting these two vertices. And then looking at the remaining vertices, there are already two
diagonals emanating from those vertices. So let's go ahead and count these.
We have to be careful here not to count them more than once. We have one here,
one here, one here, one here,
and one here. So there are a total of five
diagonals for a pentagon. So if it has five sides it has five diagonals.
Here we have a hexagon, so we'll start here at this vertex. We have one,
two, three diagonals from each vertex.
So if we go to this vertex here we'd have one,
two, and three. This is going to get much more difficult to count, so let'sgo ahead and keep
track of it. So far we have six.
From this vertex we only have one,
two more that we can sketch, so there's two more.
Again, it's helpful to remember there are three diagonals from each vertex.
Looking at this vertex there's already two sketched, so we have one more here,
so that's one more. And then the remaining vertices already have three
diagonals sketched. So when we have six sides there are
nine diagonals.
Now let's go ahead and take a look at a couple more, maybe we can see a pattern.
Here we have a polygon with seven sides, or a heptagon.
So each vertex will have one, two,
three, four diagonals emanating from it,
so there's four. And we'll have another four from this vertex,
that'll give us a total eight so far.
Now this vertex has one already sketched, so that'll be three more.
One, two, three.
This vertex has two already, so we can sketch two additional
diagonals, one here and one here.
As you can see it's getting messy. This vertex has one, two, three diagonals already, so there's one more,
that would be this one here. This one
already has one, two, three, four. And this one already has one,
two, three, four. So if there are seven sides
there are fourteen diagonals. I'm going to go ahead and stop here. You may want to investigate this further yourself.
It is a fairly challenging formula to develop from this pattern,
but the formula is, n times the quantity n minus three,
divided by two where n is a number of sides.
So we left off with the hectagon, but if we had an octagon,
with the number of diagonals for an octagon, or a polygon that has
eight sides would tell us that n is equal to eight.
So the number of diagonals would be eight times eight minus three,
divided by two. This is going to be a eight times five,
that's forty divided by two, that's twenty diagonals.
So if you have the time you may want to see if you can sketch
all twenty diagonals for the octagon.
We'll go ahead and stop here for this video. I hope you found this helpful.