Finding the Sum of the Interior Angles of a Convex Polygon

Fun with AnglesMrs. 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 – 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 - 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 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-geo-spr06/ljohnson/geolp2.doc.

Kuta Software LLC. (2011). “Introduction to Polygons” Infinite Geometry. Retrieved (04 Dec. 2011) from <http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/6-Introduction%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/interior-angles-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>