Interior and Exterior Angles
In each regular polygon, draw all the diagonals from 1 vertex.
Complete the table below using what we’ve found in our shapes above.
Polygon
# of sides (n)
# of triangles
Sum of interior
angles
Measure of ONE
interior angle
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
n-gon
n
Polygon Interior Angles Theorem
Using the table, what is the sum of the interior angles of a convex regular n-gon?
What is the measure of ONE angle in a regular n-gon?
Exterior Angles
Let’s look back at the shapes we started with and extend one segment on each shape.
What is the measure of the angle that you created by extending the segment of the shape?
Use your knowledge of the interior angle measurement and the measure of a line to
determine the measure of the exterior angle.
What do we do to one exterior angle to find the total of the exterior angles in a regular
polygon?
Shape
Triangle
Square
Pentagon
Hexagon Octagon
Dodecagon
One exterior Angle
Sum of exterior
Angles
Polygon Exterior Angle Theorem: The sum of the measure of the EXTERIOR angles of a
convex n-gon is 360°.
n-gon
Practice!!!
1. A heptagon has 4 interior angles that measure 160° each and two interior angles that are
right angles. What is the measure of the other interior angle?
2. Find the measure of each angle in a regular 11-gon.
3. The measure of each exterior angle of a regular polygon is 40°. How many sides does the
polygon have?
4. Find the value of x in the polygon.
114
105
x
135
102
5. The measure of each interior angle of a regular polygon is 165°. How many sides does
the polygon have?
6. Shown below is the floor plan of an outdoor theater stage. The stage has two right
angles and two angles that measure 135° each. The remaining interior angles are congruent.
What is the measure of each of these angles?
7. A robot walks around a museum exhibit and only makes 15˚ turns. What shape is the
exhibit the robot is walking around? (15˚ is the exterior angle of the exhibit)
8. If we’re given that the sum of the exterior angles of a polygon is 360˚, do we know what
shape the polygon is?
9. If the sum of the interior angles of a polygon measure 3,420˚, how many sides does the
polygon have.
10. Would the formulas we have created for regular convex polygons also work for concave
polygons? Why or why not?
Interior and Exterior Angles Homework
1.
The measure of each exterior angle of a regular polygon is 40˚. How many sides
does the polygon have?
2. Find the sum of the measures of the interior angles in convex 16-gon.
3. The measure of each interior angle of a regular polygon is 162˚. How many sides
does the polygon have?
4. The measure of each exterior angle of a regular polygon is 6˚. How many sides does
the polygon have?
Find the sum of the measures of the interior angles of the convex polygon.
5. 10-gon
7. 15-gon
9. 20-gon
6. 12-gon
8. 18-gon
10. 100-gon
11. A convex quadrilateral has interior angles that measure 80˚, 110˚, and 80˚. What is
the measure of the fourth interior angle?
You are given the measure of each interior angle of a regular n-gon, find
the value of n.
12. 144˚
14. 140˚
13. 120˚
15. 157.5˚
You are given the number of sides of a regular polygon find the measure
of each exterior angle.
16. 12
18. 21
17. 11
19. 15
You are given the measure of each exterior angle of a regular n-gon, find
the value of n.
20. 60˚
22. 72˚
21. 20˚
23. 10˚
24. A convex hexagon has exterior angles that measure 48˚, 52˚, 55˚, 62˚ and 68˚.
What is the measure of the exterior angle of the sixth vertex?