Finding the Sum of the Interior Angles of

Finding the Sum of the Interior Angles of a Convex Polygon Fun with Angles Mrs. Ribeiro’s Math Class

Review Terms n Polygons n Convex and Concave Polygons n 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 any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it)

Review Terms n Side n Adjacent v. Opposite n Diagonals

Review Concepts n n 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 third angle = 180° - 134° = 46°

Divide a Polygon into Triangles n n n 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 n n 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 = 156 n Expand (n-2) ⇒ 180 n - 360 = 156 n Subtract 156 n from both sides: ⇒ 180 n - 360 - 156 n = 0 Add 360 to both sides: ⇒ 180 n - 156 n = 360 Subtract 180 n-156 n ⇒ 24 n = 360 Divide by 24 ⇒ n = 360 ÷ 24 = 15

References n n 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-geospr 06/ljohnson/geolp 2. doc. Kuta Software LLC. (2011). “Introduction to Polygons” Infinite Geometry. Retrieved (04 Dec. 2011) from <http: //www. kutasoftware. com/Free. Worksheets/Geo. Worksheets/6 Introduction%20 to%20 Polygons. 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/interior-anglespolygons. html&qs=825_826_827_828_1779_829_1780> Pierce, Rod. (2010). “Interior Angles of Polygons. ” Mathsis. Fun. com. Retrieved (04 Dec. 2011) from <http: //www. mathsisfun. com/geometry/interior-angles-polygons. html>