Exterior Angles of Polygons

Show your work for each problem.

1.
What is an exterior angle of a polygon?
1. a)Half of the interior angle
2. b)The angle formed by extending one side past a vertex
3. c)The angle inside the polygon at a vertex
4. d)The angle between two diagonals
Answer: The angle formed by extending one side past a vertex
An exterior angle is formed when one side of a polygon is extended past a vertex. It's the angle between the extended side and the adjacent side, measured outside the polygon.
2.
What is the sum of all exterior angles of ANY convex polygon?
1. a)
2. b)It depends on the number of sides
3. c)
4. d)
Answer:
The sum of all exterior angles of any convex polygon is always . This is true whether the polygon has 3 sides or 100 sides!
3.
If an interior angle measures , what is the exterior angle at that vertex?
Answer: 110
Exterior angle . Interior and exterior angles at the same vertex always add up to .
4.
What is the measure of each exterior angle of an equilateral triangle?
Answer: 120
Each interior angle of an equilateral triangle is . So each exterior angle is . You can verify: ✓
5.
What is each exterior angle of a regular pentagon?
Answer: 72
Exterior angle . Each exterior angle of a regular pentagon is .
6.
A regular polygon has exterior angles of each. What type of polygon is it?
1. a)Heptagon (7 sides)
2. b)Octagon (8 sides)
3. c)Decagon (10 sides)
4. d)Nonagon (9 sides)
Answer: Nonagon (9 sides)
sides. A polygon with 9 sides is called a nonagon.
7.
What is each exterior angle of a regular decagon (10-sided polygon)?
Answer: 36
Exterior angle . As polygons get more sides, each exterior angle gets smaller.
8.
A triangle has interior angles of , , and . Find all three exterior angles and verify they sum to .
Answer: 135, 115, 110
1. What is the exterior angle at the vertex? 135
2. What is the exterior angle at the vertex? 115
3. What is the exterior angle at the vertex? 110
4. What is the sum of all exterior angles? 360
9.
Find each exterior angle of a regular octagon and its corresponding interior angle.
Answer: 45, 135
1. How many sides does an octagon have? 8
2. What is the exterior angle formula for regular polygons? 360/n
3. What is each exterior angle? 45
4. What is the corresponding interior angle? 135
10.
A regular polygon has interior angles of each. How many sides does it have?
Answer: 12
Exterior angle . Number of sides . It's a dodecagon.
11.
Which has a larger exterior angle: a regular hexagon or a regular octagon?
1. a)Octagon ()
2. b)Hexagon ()
3. c)They are equal
4. d)Cannot be determined
Answer: Hexagon ()
Hexagon: . Octagon: . The hexagon has a larger exterior angle. Fewer sides = larger exterior angles.
12.
In a quadrilateral, three exterior angles are , , and . Find the fourth exterior angle and all four interior angles.
Answer: 105, 100, 80, 105, 75
1. What is the sum of the three known exterior angles? 255
2. What is the fourth exterior angle? 105
3. What is the interior angle corresponding to exterior? 100
4. What is the interior angle corresponding to exterior? 80
5. What is the interior angle corresponding to exterior? 105
13.
A robot traces a regular polygon path, turning at each corner. How many sides does the polygon have?
Answer: 15
sides. This is a regular 15-sided polygon (pentadecagon).
14.
The exterior angle of a regular polygon is twice its interior angle. Find the number of sides.
Answer: 3
1. If interior angle is , what is the exterior angle in terms of ? 2x
2. Interior + Exterior = ? 180
3. So . What is (the interior angle)? 60
4. What is the exterior angle? 120
5. How many sides? Use 3
15.
What happens to the exterior angle as a regular polygon gets more and more sides (approaching a circle)?
1. a)Approaches
2. b)Stays at
3. c)Approaches
4. d)Approaches
Answer: Approaches
As , exterior angle . A circle has infinitely many "sides" and each exterior angle is essentially , while interior angles approach .
16.
A pentagon has exterior angles of , , , and . Find the fifth exterior angle and verify by calculating all interior angles.
Answer: 70
1. What is the sum of the four known exterior angles? 290
2. What is the fifth exterior angle? 70
3. What should be the sum of interior angles of a pentagon? 540
4. Calculate: 540