# sum of interior and exterior angles in polygons

How to determine the measure of each exterior angles for a regular hexagon
How to determine the measure of each exterior angles for a regular hexagon

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
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:
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
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°

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