1. Exploring angles of polygons
Investigation 3
Exploring angles of polygons

2. Exploring Interior Angles of Convex Polygons
Regular and/or Irregular Polygons

3. How many sides for each regular polygon?
# of
Sides
Triangles
Sum of the
Int. Angles
Measure of each angle

4. Now draw all possible diagonals, but only from one vertex of each polygon
Sides
Triangles
Sum of the
Int. Angles
Measure of each angle 3 4 5 6

5. How many triangles are formed in the interior of each regular polygon?
Sides
Triangles
Sum of the
Int. Angles
Measure of each angle 3 4 5 6

6. What is the sum of the interior angles of each regular polygon
What is the sum of the interior angles of each regular polygon? (use # of Δ & 180)
# of
Sides
Triangles
Sum of the
Int. Angles
Measure of each angle 3 1 4 2 5 6

7. Now divide the sum of the angles by the number of side for each reg
Now divide the sum of the angles by the number of side for each reg. polygon
# of
Sides
Triangles
Sum of the
Int. Angles
Measure of each angle 3 1
180° 4 2
360° 5
540° 6
720°

8. What connection can you make about the # of sides to the # of triangles?
Sum of the
Int. Angles
Measure of each angle 3 1
180°
60° 4 2
360°
90° 5
540°
108° 6
720°
120°

9. Formula for the Sum of the Interior Angles of a Convex Polygon (Regular & Irregular)
To find the sum of the interior angles of a polygon, use the formula below, where 𝑛 is the number of sides of the polygon.
𝑛−2 180°
Find the sum of interior angles of an octagon
8−2 180°
6 180°
1080°

10. Formula for the Measure of Each Interior Angle of a Regular Polygon
To find the measure of each interior angle of a regular polygon, use the formula below, where 𝑛 is the number of sides of the polygon.
𝑛−2 180° 𝑛
Find the measure of each angle of a regular pentagon
5−2 180° 5
108°

11. Exploring Exterior Angles of Convex Polygons
Regular and/or Irregular Polygons

12. Draw one exterior angle from each vertex of the regular polygons
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles

13. How many exterior angles are formed on each regular polygon?
# of
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles

14. From prev. exploration, what is the measure of each int
From prev. exploration, what is the measure of each int. angle of the reg. polygons?
# of
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles 3 4 5 6

15. By the Linear Pair Thm, what is the measure of each ext
By the Linear Pair Thm, what is the measure of each ext. angle of the reg. polygons?
# of
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles 3
60° 4
90° 5
108° 6
120°

16. Find the sum of the exterior angles of the regular polygons?
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles 3
60°
120° 4
90° 5
108°
72° 6

17. Find the sum of the exterior angles of the regular polygons?
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles 3
60°
120°
360° 4
90° 5
108°
72° 6

18. Sum of the Exterior Angles of a Convex Polygon (Regular & Irregular)
The sum of the exterior angles of any convex polygon is always 360°
Find the sum of the exterior angles of these polygons:
Dodecagon
360°
Chiliagon
n-gon

19. What would be an easier way to find the measure of each exterior angle of a reg. polygon?
Ext. Angles
Measure of each
Int. Angle
Ext. Angle
Sum of Ext. Angles 3
60°
120°
360° 4
90° 5
108°
72° 6

20. Formula for the Measure of Each Exterior Angle of a Regular Polygon
To find the measure of the exterior angle of a regular polygon, use the formula below, where 𝑛 is the number of sides of the polygon.
360° 𝑛
Find the measure of each exterior angle of a regular triangle
360° 3
120°

21. Exploring Central Angles of Convex Polygons
Regular and/or Irregular Polygons

22. Draw all radii to each vertex of the inscribed regular polygons
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle

23. How many central angles are formed for each regular polygon?
# of
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle

24. How many arcs are formed for each regular polygon?
# of
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle 3 4 5 6

25. What is the sum of all the arcs for each inscribed regular polygon?
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle 3 4 5 6

26. How could we find the measure of each central angle for regular polygons?
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle 3
360° 4 5 6

27. Sum of the Central Angles of a Convex Polygon (Regular & Irregular)
The sum of the central angles of any convex polygon is always 360°
Find the sum of the central angles of these polygons:
Dodecagon
360°
Chiliagon
n-gon

28. Find the measure of each central angle for each regular polygon?
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle 3
360° 4 5 6

29. Find the measure of each central angle for each regular polygon?
Central Angles
# of Arcs
Sum of the Arcs
𝒎 of each
Central Angle 3
360°
120° 4
90° 5
72° 6
60°

30. Formula for Central Angle Measure of a Regular Polygon
To find the measure of each central angle of a regular polygon, use the formula below, where 𝑛 is the number of sides of the polygon.
360° 𝑛
Find the measure of each central angle of a regular hexagon
360° 6
60°

31. Looking Ahead Exploring angles of polygons prepares us for:
Lesson 34: Properties of Parallelograms
Lesson 66: Finding Perimeters and Areas of Regular Polygons
While we spent the day using regular polygons, remember much of what we learned today applies to irregular polygons as well
You will need to take care when problem solving