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
What do all the exterior angles of a polygon add up to be. If we know the sum of the exterior
angles in triangle, then what can we conclude about the sum of the exterior angles in any other
polygon?
Plans to Solve/Investigate the Problem
I plan to construct a triangle, a quadrilateral, and a pentagon. From each of these constructions, I
will then measure each exterior angle and obtain the sum. I will then see if a can find a
relationship between the sum of the exterior angles and the number of sides.
Investigation/Exploration of the Problem
The first polygon that I constructed was a triangle. To construct a triangle, I created three
random points. I then connected those three points by constructing line segments between the
three points in the form of a triangle.
C
A
B
I then selected the segments CB and AB, and constructed parallel lines through those segments.
Once I constructed the lines, I placed random points on either side of Point A, Point B, and Point
C in order to be able to measure the exterior angle.
H
C
I
G
D
B
A
F
E
I then measured the interior angles by selecting the three associated points and measuring the
angle. For angle ABC, I selected Point A, Point B, and Point C and then found the angle to be
47.55 degrees. For angle ACB, I selected Point A, Point C, and Point B respectively, and then
found the measurement of the angle to be 33.33 degrees. To measure angle CAB, I selected
Point C, Point A, and Point B respectively, and found the measurement to be 99.12 degrees.
Once I found this information, I could have just simply subtracted the measure of each angle
from 180 degrees because we know that a straight line always has an angle measurement of 180
degrees. . If each interior angle and its respective exterior angle were added together, then the
measurement was 180 degrees. However, to prove this, I went ahead and measured each
exterior angle itself. I found angle FAG to be 80.88 degrees, angle HCI to be146.67 degrees, and
angle DBE to measure 132.45 degrees. I found the sum of the three exterior angles of the triangle
to be 360 degrees.
H
C
I
mABC = 47.55
mDBE = 132.45
mCAB = 99.12
mFAG = 80.88
mACB = 33.33
mHCI = 146.67
mABC+mCAB+mACB = 180.00
mDBE+mFAG+mHCI = 360.00
mABC+mCAB+mACB+mDBE+mFAG+mHCI = 540.00
G
D
F
A
B
E
The second structure that I constructed was a quadrilateral. I constructed the quadrilateral and
then measured both the interior angles and the exterior angles. I found the sum of the
measurement of the interior angles to be 360 degrees and the sum of the exterior angles to be 360
degrees also.
Q
J
M
S
T
R
P
O
U
K
L
V
mOKP = 63.99
mJKL = 116.01
mQJR = 116.61
mKJM = 63.39
mSMT = 62.96
mJML = 117.04
mULV = 116.43
mMLK = 63.57
mJKL+mKJM+mJML+mMLK = 360.00
mOKP+mQJR+mSMT+mULV = 360.00
mJKL+mKJM+mJML+mMLK+mOKP+mQJR+mSMT+mULV  = 720.00
From the results of these two polygons, one can determine the relationship between the interior
angles and the sum of the exterior angles. The sum of the interior angles for a polygon with n
number of sides is (n-2)180. In this formula, n represents the number of sides. Because an
interior angle and its corresponding exterior angle always must add up to equal 180, the sum of
all the interior and exterior angles of a polygon is the 180n, where n represents the number of
sides. Therefore, the sum of the exterior angles for all convex polygons is 360 degrees.
Sum of Interior Angles = (n-2)180
Sum of All Angles = 180n
Sum of Exterior Angles = 180n – {(n-2)180}
= 180n -180n + 360
= 360.
Author & Contact
Jennifer C. Woods
Jenn4507@yahoo.com