InterMath
Title: Sum of Angles in a Polygon
Problem Statement
What is the sum of the angles of a triangle? 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 find the sum of a triangle’s angle in a triangle, quadrilateral, and a pentagon. I am also trying to determine the sum of the angles in a convex polygon. According to the dictionary from the Intermath website, a convex polygon is a polygon that has all of its interior angles less than 180.
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M6G1
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.
Students will further develop their understanding of plane figures. a. Determine and use lines of symmetry. b. Investigate rotational symmetry, including degree of rotation. 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.
M6P1
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.
M6P3 Use the language of mathematics to express mathematical ideas precisely.
M6P4
Students will make connections among mathematical ideas and to other disciplines. a. Recognize and use connections among mathematical ideas. b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

Plans to Solve/Investigate the Problem
I plan to construct a quadrilateral, pentagon, and a hexagon in GSP. Next, I plan to create as many triangles as possible by drawing a diagonal from a single vertex to other vertices in each polygon. I’m constructing triangles because I know that the sum of the measure of the interior angles of a triangle measure 180°. Then I plan to measure the interior angles of each polygon to determine the sum of the angles in an individual polygon.
Investigation/Exploration of the Problem
I started my investigation by constructing a triangle in GSP. Then I measured the angles of the triangle. The sum of the angles totaled 180
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.
D
E
C N
A tri angl e no dia gonal
B
K
J
L quadi l ateral one di agonal
R two triangl es
Q
M
O pentagon two d iagonals three triangles
P
H G hexagon three di agonal s four tria ngles
F
Next, I constructed a quadrilateral. After I constructed the quadrilateral, I used one of the vertices and drew a diagonal to all the angles that are not adjacent to that vertex. I noticed that this diagonal created two triangles in the quadrilateral, three diagonals in the pentagon, and four diagonals in the hexagon. So I assume that the measure of the interior angles of the quadrilateral would be 2(180) or 360°, pentagon 3(180) or 540°, hexagon 4(180) or 720° and a decagon
8(180) or 1460°.
Triangle m

CAB = 90.00
 m

ABC = 48.88
 m

BCA = 41.12
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C
Trian gle ABC
B A m

BCA+m

ABC+m

CAB = 180.00
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Quadrilateral:
A quadrilateral can be divided in to two triangles which has an angle measure of 180° each, hence 180*2 or 360°.
Rectangl e DEFG
E
G m

FDE = 89.50
 m

DEG = 90.50
 m

EGF = 89.80
 m

GFD = 90.21
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D m

FDE+m

DEG+m

EGF+m

GFD = 360.00
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Pentagon
A pentagon can be divided into three distinct triangles. Therefore, the sum of the angles of a pentagon is 3*180 or 540°, so there seem to be a pattern forming. The pattern is the number of sides of a polygon minus two will give you the number of triangles in a polygon.
A pentagon has five sides. I subtracted 5-2=3. There are three distinct triangles in a pentagon.
L m

MLK = 126.22
 m

LKO = 103.73

K m

M
NML = 92.47

O m

KON = 130.20

N m

ONM = 87.39
 m

MLK+m

LKO+m

KON+m

ONM+m

NML = 540.00
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Hexagon
A Hexagon has three diagonals drawn from a single vertex. These diagonals form four triangles; therefore the sum of the angles is 4*180 or 720°.
M N
R
Q m

NMR = 117.63
 m

ONM = 124.23
 m

PON = 119.40
 m

QPO = 116.37
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O m

RQP = 117.65
 m

QRM = 124.73
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P m

NMR+m

ONM+m

PON+m

QPO+m

RQP+m

QRM = 720.00
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I noticed that a relationship between the sum of angles in a polygon can be determine by the number of diagonals from a single vertex, and the number of diagonals that can be created from a single vertex can be determined by subtracting three from the number of sides of a polygon.
After subtracting three from the number of sides of a polygon, the remaining sides will give us the number of triangles in a polygon. Then the number of triangles can be multiplied the by 180.
Example : A decagon has 10 sides, so I subtracted 3 from 10. 10-3 gives me 7 diagonals that can be created in a decagon. Seven diagonals create 8 triangles, which will be 8*180, or 1440°. The number of triangles can also be determined by subtracting 2 from the number of sides and multiplying the remaining sides by 180. 10-2=8 this is shown below in the decagon. This equation will give you the sum of the angles of any polygon.
I
J
A
B
C m

JAB = 143.57
 m

ABC = 143.81
 m

BCD = 144.50
 m

CDE = 142.11
 m

DEF = 144.53
 m

EFG = 146.89
 m

FGH = 141.34
 m

GHI = 146.06
 m

HIJ = 142.11
 m

IJA = 145.08
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H D
I multip li ed the number of tri angle s insi de the decagon times 180 degrees
180

8 = 1440.00
G E
F m

JAB+m

ABC+m

BCD+m

CDE+m

DEF+m

EFG+m

FGH+m

GHI+m

HIJ+m

IJA = 1440.00
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Extensions of the Problem
As an extension of this investigation I would have my students created a chart of the number of diagonals and sum of the angle measurements. This chart would also include an equation that would work for any polygon.
Polygons
Triangle
Number of sides
3
Number of diagonals drawn from a single vertex
0
Sum of the Angle measure (N-
2)*180
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1*180=180
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Quadrilateral 4 1 2*180=360
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Pentagon
Hexagon
Heptagon
Octagon
N-gon
5
6
7
8
N
2
3
4
5 n-3
3*180=540
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4*180=720
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5*180=900
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6*180=1080

(n-2)*180
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.
Author & Contact
Norma Smith and contact information. norma_smith19@yahoo.com
Link(s) to resources, references, lesson plans, and/or other materials http://www.geom.uiuc.edu/~dwiggins/conj07.html http://mathworld.wolfram.com/RegularPolygon.html