InterMath
Title
Diagonals in a Polygon
Problem Statement
A diagonal is a line segment that connects non-adjacent vertices in a polygon. Consider the
number of diagonals in a triangle, quadrilateral, pentagon, hexagon, heptagon, and
octagon. What pattern do you notice? Use this pattern to predict the number of diagonals in
a dodecagon (12-sided polygon).
Problem setup
I am trying to find a pattern or relationship in the number of diagonals in various polygons. I
believe there will be a connection between the number of sides of the polygon and the number of
diagonals inside the polygon.
Plans to Solve/Investigate the Problem
I will start this investigation by constructing/sketching each polygon and connecting the vertices
to form the diagonals with each polygon. I plan to make a chart in Excel to show the number of
sides that each polygon has and the number of diagonals inside the polygons. I am expecting to
find a pattern/relationship with the numbers.
Investigation/Exploration of the Problem
I began this investigation with the construction of a triangle in GSP. I tried to connect vertices to
make a diagonal but it wasn’t possible. I entered the data into the spreadsheet.
I constructed a quadrilateral in GSP. I used the vertices to construct diagonals inside the figure.
Two diagonals were formed in the quadrilateral. I entered this data into the spreadsheet.
2
I constructed a pentagon in GSP. I used the five vertices to construct diagonals inside the figure.
Five diagonals were formed in the pentagon. I entered this data into the spreadsheet.
5
I constructed a hexagon in GSP. I used the six vertices to construct diagonals inside the figure.
Nine diagonals were formed in the hexagon. I entered this data into the spreadsheet.
9
I started to notice a pattern among the numbers of diagonals. (0,2,5,9…) I believe that each
subsequent number is being added to the previous number of diagonals. I predicted the next
number would be 14.
I constructed a heptagon in GSP. I used the seven vertices to construct diagonals inside the
figure. Fourteen diagonals were formed in the heptagon. I entered this data into the spreadsheet.
14
Believing I had found a pattern, I made a prediction for a octagon. I predicted 20 diagonals.
After constructing an octagon with diagonals, my prediction proved correct.
20
Knowing that I could use this pattern to easily identify the number of diagonals inside a
dodecagon, I wanted to find a rule for the pattern so that I could calculate the number of
diagonals inside any polygon.
Using the Excel spreadsheet, I tried to find a connection between the number of sides and the
diagonals. Nothing definitive jumped out at me.
Then, I started listing patterns that I was familiar with and looked for a connection to the current
pattern. The triangular numbers pattern came to mind. 1, 3, 6, 10, 15, 21, ….
polygon
# of sides # of diagonals
triangle
3
0
quadrilateral
4
2
pentagon
5
5
hexagon
6
9
heptagon
7
14
octagon
8
20
term
1
2
3
4
5
6
triangular numbers
1
3
6
10
15
21
n
(n(n+1))/2
With these numbers listed beside my current pattern, I saw that the term in the pattern of
triangular numbers was two less than the number of sides of each polygon.
The rule/pattern for triangular numbers is n(n+1)/2. Knowing that the numbers in my diagonal
pattern are one less than each of the triangular number, I discovered a new rule/pattern.
Letting s represent the number of sides, I had to have s-2 in place of each n value. Therefore,
(s-2)(s-1)/2 – 1 is the rule for my diagonals pattern. I tested it with my chart. It was successful
on all the polygons in my chart.
With this pattern, I can now state that a 17-sided polygon would have ((s-2)(s-1))/2 - 1 diagonals
(w/ s representing the number of sides of the polygon).
For a dodecagon (12 sides), (12-2)(12-1)/2 – 1…….(10)(11)/2 – 1…….110/2 – 1…..55 -1 = 54.
polygon
# of sides
triangle
3
quadrilateral
4
pentagon
5
hexagon
6
heptagon
7
octagon
8
# of diagonals
0
2
5
9
14
20
s
(s-2)(s-1)/2 -1
12
54
Extensions of the Problem
(for my students) Is the number of diagonals the same for regular and irregular polygons? Why
or why not? Sketch to prove your answer.
GPS connections
This task does not directly correlate to a specific GPS standard although 5th and 6th grade
curriculum involves constructions of polygons (which would lend itself to constructing
diagonals).
Author & Contact
Erin Lee Hutto
Link(s) to resources, references, lesson plans, and/or other materials