Interior/Exterior Angles in Regular Polygons

R.4.G.2 Solve problems using properties of polygons:

sum of the measures of the interior angles of a polygon

interior and exterior angle measure of a regular polygon or irregular polygon

number of sides or angles of a polygon

Unit 3 Comp 4B

Regular Polygon: a polygon with all congruent sides and all congruent angles

Diagonal: a line drawn through the interior of a polygon from a vertex to a non-consecutive vertex.

Definitions

Make a drawing of the indicated polygon. Then draw all diagonals from ONE vertex.

Triangle Quadrilateral Pentagon

Hexagon Heptagon Octagon

How many triangles were formed?

Polygon Number of Sides

Number of triangles formed by drawing all

diagonals from one vertex

Sum of angle

measures in each triangle

The sum of the interior angles

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

How many triangles were formed?

Polygon Number of Sides

Number of triangles formed by drawing all

diagonals from one vertex

Sum of angle

measures in each triangle

The sum of the interior angles

Octagon

Nonagon

Decagon

N-gon

20-gon

The sum of the measures of the interior angles of an n-gon is 180(n-2)

Polygon Interior Angle Sum Theorem

Find the sum of the interior angles of the following polygons:

Dodecagon

18-gon

Examples

Draw an exterior angle at each vertex of the polygons.

Measure each exterior angle and find the sum of all the exterior angles for each polygon. Record your results in the table.

Exterior Angles of Polygons

Exterior Angles of Polygons

Polygon Number of Sides Measures of one Exterior Angle

Sum of the Measures of the Exterior

Angles

Triangle

Quadrilateral

Pentagon

Hexagon

n-gon

20-gon

The sum of the exterior angles of any polygon is

Conclusion

360 degrees!

To find one exterior angle of any polygon with “n” sides:

Interior Angle + Exterior Angle =

More Conclusions

n

360

180o

Find the measure of each interior and exterior angle of a regular octagon.

n =

Exterior angle =

Interior angle =

Example

8

135o

45o

Find the measure of each interior and exterior angle of a 22-gon.

Interior: 163.6o

Exterior: 16.4o

Examples

If the exterior angle measures 40o in a regular polygon, how many sides does it have?

More Examples!

If the exterior angle measures 120o in a regular polygon, how many sides does it have?

So Many Examples!

Last Page of Examples!

xo

xo

(x+10)o

(x+10)o(3x-17)o

2xo

xo yo

55o