Polygons

A polygon is a closed figure formed by a finite number of

segments such that:

1. the sides that have a common endpoint

are noncollinear, and

2. each side intersects exactly two other

sides, but only at their endpoints.

Polygons are named by number of sides

Number of Sides Polygon

3

4

5

6

7

8

9

10

12

n

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodecagon

n-gon

Regular Polygon

A convex polygon in which all the sides are congruent

and all the angles are congruent is called a regular

polygon.

Draw a:

Quadrilateral Pentagon

Hexagon Heptagon

Octogon

Then draw diagonals to create triangles.

A diagonal is a segment connecting two

nonadjacent vertices (don’t let segments cross)

Add up the angles in all of the triangles in

the figure to determine the sum of the

angles in the polygon.

Complete this table

Polygon # of sides # of triangles Sum of

interior angles

Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a

convex n-gon is (n – 2) • 180.

Examples –

1. Find the sum of the measures of the interior angles of a

16–gon.

2. If the sum of the measures of the interior angles of a

convex polygon is 3600°, how many sides does the

polygon have.

3. Solve for x.

4x – 2

82

108

2x + 10

(16 – 2)*180

(n – 2)*180 = 3600

180n – 360 = 3600

+ 360 + 360

180n = 3960

180 180

n = 22 sides

(4 – 2)*180 = 360

108 + 82 + 4x – 2 + 2x + 10 = 360

6x + 198 = 360

6x = 162

6 6

x = 27

= 2520°

Draw a quadrilateral and extend the sides.

There are two sets of angles formed when the

sides of a polygon are extended.

• The original angles are called interior angles.

• The angles that are adjacent to the

interior angles are called exterior angles.

These exterior angles can be formed when any

side is extended.

What do you notice about the interior angle and

the exterior angle?

What is the measure of a line?

What is the sum of an interior angle with the

exterior angle?

They form a line.

180°

180°

If you started at Point A, and

followed along the sides of

the quadrilateral making the

exterior turns that are

marked, what would happen?

You end up back where you

started or you would make a

circle.

What is the measure of the

degrees in a circle?

A

B

C

D

360°

The sum of the measures of the exterior angles of a

convex polygon, one at each vertex, is 360°.

Each exterior angle of a regular polygon is 360

n

where n is the number of sides in the polygon

Polygon Exterior Angles Theorem

54⁰

68⁰

65⁰

(3x + 13)⁰

60⁰

(4x – 12)⁰

Find the value for x.

Sum of exterior angles is 360°

(4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360

7x + 248 = 360

– 248 – 248

7x = 112

7 7

x = 12

Example

What is the sum of the exterior angles in an octagon?

What is the measure of each exterior angle in a regular

octagon?

360°

360°/8 = 45°