3.3: Angles of Polygons

What is a Polygon?

A polygon is a closed plane figure made up of three or more line segments that intersect only at their endpoints.

In a regular polygon all the sides are congruent, and all of the interior angles are congruent.

Interior angle measures of a polygon: The sum S of the interior angle measures of a polygon with n sides is

S (n 2) 1800

Concave and Convex Polygons

A polygon is convex when every line segment connecting any two vertices lies entirely inside the polygon.

A polygon is concave when at least one line segment connecting any two vertices lies outside the polygon.

Finding the Sum of Interior Angle Measures

Find the sum of the interior angle measures of the school crossing sign.

The sign is in the shape of a pentagon. It has 5 sides.

S (n 2) 1800

S (5 2) 1800

S 3 1800

S 5400

Practice

The honey comb has 6 sides

The web has 7 sides

Find the sum of the interior angle measures of the green polygon.

S (n 2) 1800

S (7 2) 1800

S 5 1800

S 9000

S (n 2) 1800

S (6 2) 1800

S 4 1800

S 7200

Finding an Interior Angle Measure of a Polygon

Step 1: The polygon has 7 sides. Find the sum of the interior angle measures.

Step 2: Write and solve an equation to find x.

Find the value of x.

S (7 2) 1800

S 5 1800

S 9000

140145115 120130128 x 900778 x 900

x 1220

Practice

Find the value of x.

S (6 2) 1800

S 4 1800

S 7200

S (4 2) 1800

S 2 1800

S 3600

S (5 2) 1800

S 3 1800

S 5400

135110 125120125 x 720615 x 720

x 1050

9080115 x 360285 x 360

x 750

145145110 2x 2x 540400 4x 5404x 140

x 350

Real Life Application

Step 1: A hexagon has 6 sides. Find the sum of the interior angle measures.

Step 2: Divide the sum by the number of interior angles, 6.

A cloud system discovered on Saturn is in the approximate shape of a regular hexagon. Find the measure of each interior angle of the hexagon.

S (n 2) 1800

S (6 2) 1800

S 4 1800

S 7200

7200 6 1200

Exterior Angle Measures of a Polygon

The sum of the measures of the exterior angles of a convex polygon is 3600.

Algebra w+x+y+z=3600

Finding Exterior Angle Measures

Write and solve an equation for z

Write and solve an equation for x

Find the measures of the exterior angles of each polygon.

x 5012791360x 268 360

x 920

124 z z 26 3602z 150 3602z 210

z 1050

The exterior angles measures are 1050 and (105+26)=1310

Practice

Find the measures of the interior angles of the polygon.

Write and solve an equation to find x.

909090 x x 3602702x 3602x 90

x 450

The exterior angle measures are 900, 450, 900, 900, and 450.