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

Review Terms Polygons Convex and Concave Polygons 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 Side Adjacent v. Opposite Diagonals

Review Concepts 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 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 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 Use pencil and paper – work with a shoulder partner

Sum of interior angles of a quadrilateral: 360° Given angles sum = 113° + 51° + 82° = 246° Fourth angle Find an interior angle a + b + c + d = 360º a + b + c = 246º 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 = 156n Subtract 156n from both sides: ⇒ 180n n = 0 Add 360 to both sides: ⇒ 180n - 156n = 360 Subtract 180n-156n ⇒ 24n = 360 Divide by 24 ⇒ n = 360 ÷ 24 = 15

References Johnson, Lauren. (27 April 2006). “Polygons and their interior angles.” University of Georgia. Retrieved (04 Dec. 2011) from spr06/ljohnson/geolp2.doc. spr06/ljohnson/geolp2.doc. Kuta Software LLC. (2011). “Introduction to Polygons” Infinite Geometry. Retrieved (04 Dec. 2011) from Mathopolis.com (2011) “Question 1780 by lesbillgates.” Retrieved (0 Dec. 2011) from Pierce, Rod. (2010). “Interior Angles of Polygons.” MathsisFun.com. Retrieved (04 Dec. 2011) from