InterMath | Workshop Support | Write Up Template
Title:
Sum of exterior angles
Problem Statement:
What is the sum of the exterior angles in a triangle? In a quadrilateral? In a pentagon? What is true about the
sum of the exterior angles in any convex polygon? Why?
Problem setup:
I will determine the sum of the exterior angles of any triangle, quadrilateral, and pentagon. I will investigate to
see why is this relationship holds true. I will measure the exterior angles of several different convex polygons –
triangle, quadrilateral, pentagon, hexagon, and octagon – and calculate the sum of the exterior angles of each.
Plans to Solve/Investigate the Problem:
Using GSP I will construct a triangle, quadrilateral, and a pentagon. I will then examine the sums of the
exterior angles of each shape. I will further investigate the sum of the exterior angles of other convex polygons
– a hexagon and octagon. Finally, I will try to determine why the sums of the exterior angles measure up to 360
degrees each time.
Investigation/Exploration of the Problem :
Using Geometer’s Sketchpad (GSP), I constructed convex shapes. I started with a random triangle. I placed
three random points and then joined them with rays, so that I could see the exterior angles. On the rays, I
placed points outside of the triangle vertices so that I could have three points to measure the exterior angles.
E
mABC = 100.30
mDAE = 135.25
mBDF = 124.45
mABC+mDAE+mBDF = 360.00
A
Sum of the triangle’s exterior angles = 360
D
B
C
F
Next, I constructed a random quadrilateral. I placed four random points and connected them with rays, so that I
could extend past each vertex. By going further than the vertices, I was able to easily measure each exterior
angle of the quadrilateral.
G
H
N
I
M
K
J
mGHI = 88.53
mJIK = 99.70
mLKM = 80.30
mNMH = 91.47
mGHI+mJIK+mLKM+mNMH = 360.00
Sum of the quadrilateral’s exterior angles = 360
Then, I constructed a random pentagon. I placed five random points and connected them with rays. Again, by
connecting them with rays I was able to extend past each vertex and easily measure the exterior angles.
P
Q
Y
X
R
S
W
V
T
mPQR = 85.78
mSRT = 72.19
mUTV = 65.70
mWVX = 64.50
mYXQ = 71.83
mPQR+mSRT+mUTV+mWVX+mYXQ = 360.00
U
Sum of pentagon’s exterior angles = 360
I started to see a pattern develop. No matter how many sides the figure has, the sum of the exterior angles is
always equal to 360. I decided to look at a couple more figures – a hexagon and an octagon.
M2
L2
I2
mI 2J2K2 = 84.86
N2
K2
mL2K2M2 = 40.88
O2
J2
mN2M2O2 = 41.79
P2
T2
Q2
S2
R2
mP2O2Q2 = 90.17
mR2Q2S2 = 47.44
mT2S2J2 = 54.88
mI 2J2K2+mL2K2M2+mN2M2O2+mP2O2Q2+mR2Q2S2+mT2S2J2 = 360.00
Sum of hexagon’s exterior angles = 360
U2
X2
W2
mU2V2W 2 = 34.59
mX2W 2Y2 = 33.04
J3
V2
mZ2Y2A3 = 63.95
Y2
I3
mB3A3C3 = 62.79
Z2
mD3C3E3 = 21.41
H3
A3
B3
mF3E3G3 = 31.08
mH3G3I 3 = 59.78
G3
C3
E3
F3
mJ3I 3V2 = 53.36
mU2V2W 2+mX2W 2Y2+mZ2Y2A3+mB3A3C3+mD3C3E3+mF3E3G3+mH3G3I 3+mJ3I 3V2 = 360.00
D3
Sum of octagon’s exterior angles = 360
As I have previously shown with the triangle, quadrilateral, pentagon, hexagon, and octagon, regardless of the
number of sides a convex polygon has, the sum of the exterior angles is always 360. I further investigated by
constructing triangles out of the exterior angles of the triangle and quadrilateral. I placed each of the triangles
together and it formed a circle, therefore, further showing the relationship of the sum of exterior angles
measuring to 360 degrees.
D4
G1
U1
F
S
J1
H1
L1
C4
K1
Y3
N3
W3
E4 R 3M3
S3
O3
X3
T1
T3
This is a wonderful visual representation. This will help students to see the relationship between the sum of the
exterior angles, 360 degrees, and the degrees in a circle, 360 degrees.
Extensions of the Problem:
I decided to take a look at a concave polygon to see if this relationship continues to hold true. I constructed a
concave pentagon and measured the exterior angles.
X4
V4
Y4
U4
C5
W4
mS4T4U4 = 91.36
A5
S4
T4
Z4
mV4U4W 4 = 120.06
mX4Y4Z4 = 123.39
mC5W 4U4 = 109.51
B5
mB5Z4T4 = 95.67
mS4T4U4+mV4U4W 4+mX4Y4Z4+mC5W 4U4+mB5Z4T4 = 540.00
The sum of the exterior angles did not equal up to 360 degrees this time. The sum of the concave pentagon’s
exterior angles actually equals the sum of the interior angles of a convex and concave pentagon. Four of the
angles are actual exterior angles, however, the fifth angle is not capable of having an exterior angle. An exterior
angle must be an extension from the side of a polygon, which forms a linear pair with the interior angle. As
shown in the picture directly above, the extension of the concave angle goes inside of the polygon, making it an
interior angle.
GPS:
Geometry
Students will further develop their understanding of plane and solid geometric figures, incorporating the
use of appropriate technology and using this knowledge to solve authentic problems.
M6G1. Students will further develop their understanding of plane figures.
c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar
plane figures.
d. Interpret and sketch simple scale drawings.
e. Solve problems involving scale drawings.
M8P1 Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
Author & Contact:
Mandie Pulley
mandie_pulley@yahoo.com
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