Finding the Sum of the Interior
Angles of a Convex Polygon
Fun with Angles
Mrs. Ribeiro’s Math Class
Review Terms
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Polygons
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Convex and Concave Polygons
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Vertex (pl. Vertices)
Polygons
A plane shape (two-dimensional) with
straight sides.
Examples: triangles, rectangles and
pentagons.
Note: a circle is not a polygon because
it has a curved side
Types of Polygons
Convex Polygon
A convex polygon has no angles pointing
inwards. More precisely, no internal
angles can be more than 180°.
Concave Polygon
If there are any internal angles greater than
180° then it is concave.
(Think: concave has a "cave" in it)
Review Terms
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Side
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Adjacent v. Opposite
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Diagonals
Review Concepts
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What is the sum of the interior angles
of a triangle?
How can we use this to find missing
angles in a triangle?
a + b + c = 180º
Triangle Sum Theorem
What is the measure of the third angle?
a + b + c = 180º
Triangle Sum Theorem
The measure of the third angle is:
The interior angles of a triangle add to 180°
The sum of the given angles = 29° + 105° = 134°
Therefore the third angle = 180° - 134° = 46°
Divide a Polygon into Triangles
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Choose a vertex
Draw a diagonal to the closest vertex at
left that is not adjacent
Repeat for additional diagonals until
you reach the adjacent at right
Polygons into Triangles
Hexagon:
Quadrilateral:
Polygons into Triangles
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Let’s count triangles!…
Hexagon:
Quadrilateral
Rule for Convex Polygons
Sum of Internal Angles = (n-2) × 180°
Measure of any Angle in Regular Polygon
= (n-2) × 180° / n
Example: A Regular Decagon
Sum of Internal Angles = (n-2) × 180°
(10-2)×180° = 8×180° = 1440°
Each internal angle (regular polygon)
= 1440°/10 = 144°
Find an interior angle
What is the fourth interior angle of this
quadrilateral?
A 134°
B 129°
C 124°
D 114°
Use pencil and paper – work with a shoulder partner
Find an interior angle
Sum of interior angles of a quadrilateral: 360°
a + b + c + d = 360º
Given angles sum = 113° + 51° + 82° = 246°
a + b + c = 246º
Fourth angle
d = 360 º - 246º = 114 º
Working “Backwards”
Each of the interior angles of a regular
polygon is 156°. How many sides does this
polygon have?
A 15
B 16
C 17
D 18
Working “Backwards”
Use the formula for one angle of a regular n-sided polygon.
We know one angle = 156°
Now we solve for "n":
Multiply both sides by n ⇒ (n - 2) × 180 = 156n
Expand (n-2) ⇒ 180n - 360 = 156n
Subtract 156n from both sides: ⇒ 180n - 360 - 156n = 0
Add 360 to both sides: ⇒ 180n - 156n = 360
Subtract 180n-156n ⇒ 24n = 360
Divide by 24 ⇒ n = 360 ÷ 24 = 15
References
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Johnson, Lauren. (27 April 2006). “Polygons and their interior angles.” University of Georgia.
Retrieved (04 Dec. 2011) from http://intermath.coe.uga.edu/tweb/gcsu-geospr06/ljohnson/geolp2.doc.
Kuta Software LLC. (2011). “Introduction to Polygons” Infinite Geometry. Retrieved (04
Dec. 2011) from <http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/6Introduction%20to%20Polygons.pdf>
Mathopolis.com (2011) “Question 1780 by lesbillgates.” Retrieved (0 Dec. 2011) from
<http://www.mathopolis.com/questions/q.php?id=1780&site=1&ref=/geometry/interiorangles-polygons.html&qs=825_826_827_828_1779_829_1780>
Pierce, Rod. (2010). “Interior Angles of Polygons.” MathsisFun.com. Retrieved (04 Dec.
2011) from <http://www.mathsisfun.com/geometry/interior-angles-polygons.html>