Sum of the Interior Angles
Problem Statement
Is the sum of the angles in certain convex polygons related to the number of sides the polygon has? A non convex polygon is a polygon that has all of its interior angles more than 180. is of a quadrilateral? of a pentagon? of a hexagon? What is the sum of the angles in convex polygons in terms of the number of sides? Move the mouse pointer over the picture for a hint.
Problem setup
I am trying to establish the sum of the interior angles of a triangle, quadrilateral, pentagon, and a hexagon. I am also trying to draw a conclusion about the relationship of the sum of the angles in a convex polygon in terms of the number of sides. I predict that the sum of the angles of each polygon will increase as the number of sides increase by at least 60 degrees.
Plans to Solve/Investigate the Problem
Using GSP, I will construct random polygons and then compare the sum of the interior angles of a triangle, quadrilateral, pentagon, and a hexagon. Then I will look for the relationship in terms of the number of sides. I will also use the internet to investigate other resources
Investigation/Exploration of the Problem
According to the triangle sum conjecture, the sum of the interior angles in any triangle is
180 degrees. I constructed a triangle in GSP and measured the sum of the angles. Using sum of the interior angles in a triangle as a basis, I divided a quadrilateral into two triangles in GSP. I then measured the sum of all the angles and they measured 360 degree. Next, I constructed a pentagon and divided it into three triangles. I measured all the angles and found the sum to be 540 degrees. Finally, I constructed a hexagon and divided it into four triangles. I then measured the sum of the angles and found it to be
720 degrees. As the number of sides increased, the measure of the sum of the angles increased by 180 degrees.
m
ABC = 72.54
m
BCA = 58.59
m
CAB = 48.87
m
CAB+m
BCA+m
ABC = 180.00
B m DFG+m m m m
DFG = 44.20
GDF = 36.25
FGD = 99.54
GDF+m
FGD = 180.00
m
DEF = 95.17
m EFD = 36.50
m
FDE = 48.33
m
FDE+m
EFD+m
DEF = 180.00
m
FDE+m
EFD+m
DEF
DFG+m
GDF+m
FGD
= 360.00
G
D
F
C A
E
I
H m HJK = 70.05
m
JKH = 52.75
m
KHJ = 57.20
m
KHJ+m
JKH+m
HJK = 180.00
m HKL = 31.95
m
KLH = 129.37
m
LHK = 18.68
m
LHK+m
KLH+m
HKL = 180.00
L m
HIJ = 134.07
m JHI = 17.93
m
IJH = 28.00
m
HIJ+m
JHI+m
IJH = 180.00
m
HIJ+m
JHI+m
IJH
+ m
LHK+m
KLH+m
HKL
+ m
KHJ+m
JKH+m
HJK
= 540.00
K
J
M
O
N
P m
MNO = 30.58
m
NOM = 26.49
m
OMN = 122.93
m
OMN+m
NOM+m
MNO = 180.00
m
NOP = 91.27
Q m
OPN = 61.48
m
PNO = 27.26
m
PNO+m
OPN+m
NOP = 180.00
R m
NPQ = 64.92
m
PQN = 86.84
m
QNP = 28.24
m
QNP+m
PQN+m
NPQ = 180.00
m
NQR = 58.94
m
QRN = 92.78
m
RNQ = 28.29
m
QRN+m
RNQ+m
NQR = 180.00
m
QRN+m
RNQ+m
NQR
+ m
QNP+m
PQN+m
NPQ
+ m
PNO+m
OPN+m
NOP
+ m
OMN+m
NOM+m
MNO
= 720.00
An algebraic formula can also be used to describe this relationship.
Polygon (n-2) x 180 Sum of Interior Angles
Triangle
Quadrilateral
Pentagon
Hexagon
(3-2) x 180
(4-2) x 180
(5-2) x 180
(6-2) x 180
180
360
540
720
Extensions of the Problem
Investigate the relationship between the number of sides of a polygon and the number of diagonals. I did not see an apparent relationship between the number of sides and diagonals, however, through internet investigations I found that the diagonals is equal to
Number of sides x (number of sides – three) divides by 2.
4 si des
2 diago nals
3 si des
No di agonal s
G
B
F
A
D
E
C
H
I
L
5 si des
5 diago nals
J
K
P
O
6 si des
9 diago nals
M
Q
N
Author & Contact
Chantel Lewis chantel_lewis@putnam.k12.ga.us
R
GPS Connection: Students must understand angles and the sum of angles in a polygon.
GPS:
M5A1b. Investigate simple algebraic expressions by substituting numbers for the unknown.
M5P2. Students will investigate, develop, and evaluate mathematical arguments.
M5P3. Students will use the language of mathematics to express ideas precisely.
Link(s) to resources, references, lesson plans, and/or other materials
*www.geom.uiuc.edu/~dwiggins/conj07.html
*www.tcc.edu/vml/Lessons/jjoyner/Pagel/Formula.html
