Hi and welcome to this video about polygons!
In this video we will explore four things:
1, what polygons are;
2, the different parts of a polygon;
3, ways to classify polygons;
and 4, how to determine the number of diagonals in a polygon.
The term polygon is derived from the Greek words "polys" meaning “many” and "gonia" meaning “angle,”
so polygons have many angles.
First, let’s explore how polygons are constructed.
Consider the geometric point, represented by a dot:
The point is 0-dimensional—it has no length, width, height, nothing.
Now, let’s consider two connected points:
This is called a line, or a line segment.
A line segment is 1-dimensional; it has length, but no width or height.
When multiple line segments are connected end-to-end, polygons, such as this, can be formed:
Polygons are 2-dimensional.
They have no thickness, like this.
In order to be a polygon, the shape must be closed.
In other words, every endpoint must be connected to another endpoint.
This figure
is comprised of connected segments, but the result is not a polygon.
This shape
is closed, but not made up of any connected segments, so it is also not a polygon.
A polygon is defined as a 2-dimensional closed shape composed of line segments.
The sides of polygons are called edges and the angles created where the edges intersect are called vertices.
Polygons also have as many edges as vertices.
Polygons are named by the number of edges they have.
This polygon has 3 edges, 3 vertices, and is called a triangle.
The triangle has the smallest number of edges and vertices of any polygon—
it is impossible to create a 2-sided polygon.
Some common polygons are quadrilaterals, (which have 4 sides) pentagons, (which have 5 sides)
hexagons, (6 sides) heptagons, (7 sides) octagons, (8 sides) nonagons, (9 sides) decagons, (10 sides)
and dodecagons (which have 12 sides).
Though polygons with any number of edges have names,
the general n-gon is typically used for all other polygons, where n represents the number of sides.
For instance, a 30-sided polygon is called a triacontagon, but it’s often simply called a 30-gon.
Polygons can be regular or irregular.
Regular polygons have congruent edges and congruent vertices.
For example, this is a regular quadrilateral.
The edges are the same length and the vertices have the same measure.
This is an irregular quadrilateral.
The vertices have the same measure, but the edges have different lengths.
This is an irregular quadrilateral.
The edges are the same length, but the vertices have different measures.
This is an irregular quadrilateral.
Neither the edges nor the vertices have the same measure.
Polygons can also be convex or concave.
When one or more vertices of a polygon measures more than 180 degrees, the result is a concave polygon.
For example:
This is a Convex hexagon.
All of the vertices measure less than 180 degrees.
This is a Concave hexagon.
One vertex measures more than 180 degrees.
Now remember: only polygons with 4 or more sides can be concave because it’s not possible for a triangle
to contain an angle measuring more than 180 degrees, which is a straight line.
Concave polygons cannot be regular because all the vertices will never be the same measure.
Polygons also contain diagonals.
Diagonals are line segments joining two vertices that are not next to each other.
As you can see here, this irregular convex pentagon has 5 diagonals.
This is an irregular concave pentagon.
It also has 5 diagonals, even though the concavity causes diagonals to lie outside the polygon.
Triangles do not have diagonals because there is no way to connect two vertices with segments
that are not edges.
Now that you know the basics of polygons, let’s use the diagrams to figure out
a formula for finding the number of diagonals in any polygon.
In this case, the number of diagonals connecting to each vertex is 2,
which is 3 less than the number of vertices, 5;
2 diagonals connect to each vertex.
This is true of all diagonals of all polygons.
A nonagon has 9 minus 3, equal to 6 diagonals connected to each vertex.
The number 3 is not arbitrary here—from any vertex, diagonals cannot connect to the vertex itself
or to the vertices that they are “1" away from, because it would be edges.
That makes 3 vertices from every vertex that aren’t included.
To generalize this, we’ll use n-3, where n is the number of vertices of the polygon.
Now, each vertex has the same number of diagonals connecting to it, so in this case, we can see that the
total number of diagonal connections to vertices is 5 times, 5 minus 3, which is equal to 10.
In general, we can say the total number of diagonal connections is n times, n minus 3.
When we figure out this total, though, we are counting each diagonal twice, because diagonals have
2 endpoints.
In order to figure out the number of unique diagonals, we need to divide our total by 2.
In our pentagon, this looks like:
5
(which corresponds to our diagram).
Therefore, for any sized polygon our equation can be written as this:
n times parenthesis, n minus 3, all over 2 (or divided by 2).
Conceptually, this can be remembered as:
Using our formula, we can determine the number of unique diagonals in, for example, a 17-gon:
Which is equal to, 119 unique diagonals.
We can also see algebraically that triangles have no diagonals:
We can also figure out how many edges, or vertices, a polygon has by the number of unique diagonals.
Suppose a polygon has 44 unique diagonals.
How many edges does the polygon have?
Alright, so here we are at our answer, n (the number of edges the polygon has) is either equal to
11 or -8.
Because what we want to do is we want to make sure that we end with a result of 0,
which means that n, somewhere, has to be 11 or -8 in order to get us multiplying 0 by another answer,
resulting in 0.
Since n can’t be negative (because we can’t have negative edges) it can’t be -8,
so we know that our answer is 11.
The number of edges is 11, making it an 11-gon.
Alright, excellent!
I hope this video helped increase your knowledge of polygons and made some of their properties
a little bit clearer!
Thanks for watching, see you next time!