Number of Diagonals in a Polygon
1. Draw a triangle, a quadrilateral, a pentagon (5 sides), a hexagon (6
sides), and a heptagon (seven sides).
No Diagonals
2. A diagonal is defined to be a segment what joins and 2 non-adjacent
vertices in a polygon. Draw all the possible diagonals in each of the
above figures.
See above
3. Complete the table below:
Number of sides
Total number of
diagonals
3
0
4
2
5
5
6
7
9
14
4. Write the total number of diagonals as a function of the number of
sides.
F(x) = x(x-3)/2
5. Write that quadratic above in standard form.
F(x) = x(x-3)/2
F(x) = (x^2 -3x)/2
F(x) = ½ x^2 – (3/2)x
6. If a polygon has 77 diagonals, find out how many sides it must have using
A) logic,
A 3-sided figure (triangle) has no diagonals
A 5-sided figure (triangle) has 5 diagonals
A 10-sided figure (triangle) has 35 diagonals
A 13-sided figure (triangle) has 65 diagonals
A 15-sided figure (triangle) has 90 diagonals
Therefore, if a figure has 77 diagonals, it must have 14 sides.
B) factoring and
77 = x(x-3)/2
154 = x^2 -3x
x^2 – 3x – 154 = 0
(x-14)(x+11) = 0
x = 14; x = -11
14 sides (can’t have a negative # of sides)
C) graphing.
Polygon Diagonals
# of diagonals
100
80
60
Series1
40
20
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14
# of sides
sides
1
2
3
4
5
6
7
8
9
10
11
12
13
14
diagonals
0
0
0
2
5
9
14
20
27
35
44
54
65
77
7. Compare and contrast the ease of using each method.
Knowing the equation and factoring the quadratic is the easiest way to
determine the number of diagonals. If the number of sides of the polygon
was quite large, it would be difficult to use logic because you would have to
solve more equations than necessary. When using graphs, you would have to
solve equations for each of the polygons in order to plot their points.