The shape below consists of line segments defined by the points , , , , , and These line segments form a loop.
A chain of straight line segments that form a loop is called a polygon.
The line segments are the sides of the polygon, and the endpoints of the sides are the vertices.
For this polygon:
The vertices are , , , , , and
The sides are , , , , , and
To denote a particular polygon, we start at one vertex and loop around the polygon, reading off the vertices in order. So, the polygon above can be denoted
The singular form of the word vertices is vertex. So, is a vertex, is a vertex, etc.
The name of a polygon is determined by the number of sides it has.
The polygon below has sides. A six-sided polygon is called a hexagon.
The names of the polygons with up to ten sides are given below:
| Number of sides | Name |
|---|---|
| Triangle | |
| Quadrilateral | |
| Pentagon | |
| Hexagon | |
| Heptagon | |
| Octagon | |
| Nonagon | |
| Decagon |
What type of polygon is the figure shown below?
The given figure is a polygon with sides: and
Because it has sides, it is a heptagon.
What type of polygon is the figure shown above?
|
a
|
A decagon |
|
b
|
A hexagon |
|
c
|
A pentagon |
|
d
|
A nonagon |
|
e
|
An octagon |
What type of polygon is the figure shown above?
|
a
|
An octagon |
|
b
|
A pentagon |
|
c
|
A hexagon |
|
d
|
A heptagon |
|
e
|
The figure is not a polygon |
We can categorize polygons in different ways.
A polygon is simple if its boundary does not intersect itself. Otherwise, the polygon is complex.
A polygon is convex if it has no angles pointing inwards. Otherwise, if the polygon has angles pointing inwards, we call it concave.
It might help to remember that a concave polygon looks a bit like the entrance to a cave when viewed from the outside.
Note: A more technical way to define a convex polygon is to say that all of its internal angles are less than If you're not sure what an internal angle is yet, don't worry! We'll learn more about the internal angles of a polygon in future lessons.
Is the polygon shown above (a) simple or complex, and (b) convex or concave?
A polygon is convex if it has no internal angle larger than In plain words, a convex polygon has no angles pointing inwards. Otherwise, the polygon is called concave.
A polygon is simple if its boundary does not intersect itself. Otherwise, the polygon is called complex.
In our case, the polygon is
since the boundary doesn't intersect itself, and
since it has an angle pointing inwards.
Which of the above polygons are complex?
|
a
|
III only |
|
b
|
II and III only |
|
c
|
II only |
|
d
|
I and II only |
|
e
|
I only |
Construct the correct description for the above polygon.
|
a
|
|
|
b
|
|
|
c
|
|
|
d
|
|
|
e
|
Consider the pentagon shown below.
We have the following definitions:
Two sides are adjacent (or consecutive) if they have a common vertex.
For example, is adjacent to and
Two vertices are adjacent if there is a side joining them.
For example, the vertex is adjacent to the vertices and .
A diagonal is a line segment that joins two non-adjacent vertices.
For example, and are diagonals of our pentagon.
Which vertex/vertices are adjacent to
is the endpoint of the sides and Therefore, the vertices and are adjacent to
Which vertices are adjacent to $E?$
|
a
|
$D$ and $F$ |
|
b
|
$D$ and $E$ |
|
c
|
$D$ and $G$ |
|
d
|
$E$ and $G$ |
|
e
|
$A$ and $C$ |
Which sides are consecutive to $\overline{DE}?$
|
a
|
$\overline{AB}$ |
|
b
|
$\overline{EF}$ and $\overline{AF}$ |
|
c
|
$\overline{BC}$ and $\overline{CD}$ |
|
d
|
$\overline{AF}$ and $\overline{BC}$ |
|
e
|
$\overline{CD}$ and $\overline{EF}$ |
What are the elements and for the polygon below?
The segment joins two non-adjacent vertices of the polygon. Therefore, is a diagonal.
The point is an endpoint of the sides and Hence, is a vertex.
In the polygon shown above, what diagonal is missing?
|
a
|
$\overline{BD}$ |
|
b
|
$\overline{CD}$ |
|
c
|
$\overline{AB}$ |
|
d
|
$\overline{CE}$ |
|
e
|
$\overline{BE}$ |
What are the elements $\overline{AG}$ and $E$ for the polygon above?
|
a
|
$\overline{AG}$ is a diagonal and $E$ is a vertex |
|
b
|
$\overline{AG}$ and $E$ are both sides |
|
c
|
$\overline{AG}$ and $E$ are both diagonals |
|
d
|
$\overline{AG}$ and $E$ are both vertices |
|
e
|
$\overline{AG}$ is a side and $E$ is a vertex |
What are the elements $\overline{CD},\, \overline{AG}$ and $F$ for the polygon above?
|
a
|
$\overline{CD}$ and $\overline{AG}$ are sides, and $F$ is a vertex |
|
b
|
$\overline{CD},\, \overline{AG}$ and $F$ are sides |
|
c
|
$\overline{CD}$ and $\overline{AG}$ are diagonals, and $F$ is a vertex |
|
d
|
$\overline{CD}$ is a diagonal, $\overline{AG}$ is a side and $F$ is a vertex |
|
e
|
$\overline{CD}$ is a side, $\overline{AG}$ is a diagonal and $F$ is a vertex |