Exterior Angles of a Polygon Worksheet will introduce you to Exterior Angles Theorem and help you how to find exterior angles of a regular polygon. The Practice Page on Exterior Angles of a Polygon will help you to apply knowledge of exterior angles while solving questions in your homework. Using Exterior Angles of Polygon Worksheet with Answers you will be able to track measure your progress and achieve their best. Students can build confidence with the personalized learning available for each of them and can start learning at their own pace.

Do Check:

- Worksheet on Interior Angles of a Polygon
- Sum of the Interior Angles of an n-sided Polygon
- Sum of the Exterior Angles of an n-sided Polygon

## Questions on Exterior Angles of a Polygon

**I. **How many sides does a regular polygon have if the measure of an exterior angle is

i.30

ii. 60?

**Solution:**

i.Given that,

The measure of an exterior angle=30

The sum of all the exterior angles of the regular polygon = 360°

Let the number of sides be = n.

we know that number of sides = Sum of exterior angles / each exterior angle

= 360° / 30°= 12

Thus the regular polygon has 12 sides.

ii. Given that,

The measure of an exterior angle=60

The sum of all the exterior angles of the regular polygon = 360°

Let the number of sides be = n.

we know that number of sides = Sum of exterior angles / each exterior angle

= 360° / 60°= 6

Thus the regular polygon has 6 sides.

**II. **The ratio of the exterior and interior angle of a regular polygon is 2: 9. Find the number of sides in the polygon.

**Solution:**

Given that,

The ratio of the exterior and interior angle of a regular polygon is 2: 9

Each interior angle of a regular polygon =(n-2)180^{0}/n

where n = number of sides of the polygon

Each exterior angle of a regular polygon =360^{0}/n

360^{0}/n : (n-2)180^{0}/n = 2 : 9

360^{0}/(n-2)180^{0}=2/9

2/n-2=2/9

18=2(n-2)

10=2n-4

2n=14

n=14/2=7

Therefore, The polygon has 7 sides.

**III. **What is the measure of one exterior angle of a regular hexagon (six-sided polygon)?

**Solution :**

In any polygon, the sum of all exterior angles is 360°.

The given hexagon is a regular polygon.

So, all its exterior angles are of the same measure.

Because hexagon is a six-sided polygon, the measure of each exterior angle is= 360° / 6= 60°

So, the measure of each exterior angle of a regular hexagon is 60°.

**IV. **what is the no. of sides in a regular polygon if the interior angle is equal to its exterior angle?

**Solution:**

Let interior angle = exterior angle = x°

we know that interior angle. + exterior angle = 180°.

So. x+x=180°.

2.x=180

x = 180°/2= 90°.

But exterior angle =360°/n

90°=360°/n

n= 360°/90°= 4.

Number of sides in a regular polygon = 4.

**V. **Each interior angle of a polygon is nine times the exterior angle of the polygon. Find the number of sides.

**Solution:**

The Sum of an interior angle and exterior angle of a polygon is 180 degrees.

Let the exterior angle be x.

Interior angle is 9x.

x + 9x = 180°

10x = 180°

x= 180°/10

x=18°

The Sum of all exterior angles of a polygon is 360 degree

So n X x = 360°

n = 360°/18°

n=20

Therefore, The polygon has 20 sides.

**VI. **The sum of all the interior angles of a polygon is two times the sum of its exterior angles.

i. Find the number of sides in the polygon. Also,

ii. find the measure of each exterior angle and each interior angle.

**Solution:**

Given that,

The sum of all the interior angles of a polygon is two times the sum of its exterior angles.

i. The sum of the exterior angles of a polygon is always equal to 360^{0}.

The sum of the interior angles of polygon = 180(n−2)

= 180(n−2)=2×360

=180(n-2)=720

=n-2=720/180

= n−2=4

= n=6

Number of sides in the polygon = 6.

ii. The interior angle of a regular polygon is (n-2) 180/n

By substituting n=6, we get

=(6-2)180/10

=(4)180/10

=72^{0}

Exterior angle=180 – interior angle

= 180 – 72

=108^{0}

Therefore, the polygon has six sides interior angle is 72^{0} and the exterior angle is 108^{0}.

**VII. **The exterior angles of a hexagon are in the ratio 1: 2: 3: 4: 5: 5. Find all the interior angles of the hexagon?

**Solution:**

Given that,

The exterior angles of a hexagon are in the ratio 1: 2: 3: 4: 5: 5.

Exterior angle sum property of a polygon: 360 degrees

Therefore , x+2x+3x+4x+5x+5x=360

20x=360

x=360/20

x=18

The exterior angles are 18,36,54,72,90,90.

In any polygon, the sum of an interior angle and the sum of an exterior angle is 180.

Therefore, The interior angles are:

180-18=162

180-36=144

180-54=126

180-72=108

180-90=90

By adding them, we get,

162+144+126+108+2(90)=720= interior angle sum property of hexagon

Therefore, The interior angles of the hexagon are 162^{0},144^{0},126^{0},108^{0},90^{0}.

**VIII. **What is the measure of each exterior angle of a regular nonagon (nine-sided polygon)?

**Solution:**

In any polygon, the sum of all exterior angles is 360°.

The given nonagon is a regular polygon.

So, all its exterior angles are of the same measure.

Because nonagon is a nine-sided polygon, the measure of each exterior angle is

= 360° / 9= 40°

The measure of each exterior angle of a regular nonagon is 40°.

**IX. **The ratio between an exterior angle and the interior angle of a regular polygon is 2: 7. Find:

(a) the measure of each exterior angle.

(b) the measure of each interior angle.

(c) the number of sides in the polygon.

**Solution:**

Given that,

The ratio between an exterior angle and the interior angle of a regular polygon is 2: 7.

We know that measure of an interior angle = (n – 2)(180/n) and the measure of an exterior angle = (360/n).

2/7 = (360/n) / (n – 2)(180/n)

2/7 = (360/n) / n/(n – 2) * 180

2/7 = (360/n) / n(180n – 360)

2/7 = (360)/(180(n – 2))

2/7 = 2/(n – 2)

2(n – 2) = 7 * 2

2n – 4 = 14

2n = 18

n=18/2=9.

Therefore the number of sides in the polygon = 9.

(1) Therefore the measure of each exterior angle = 360/(n)

= 360/9

= 40^{0}.

(2) Therefore the measure of each interior angle = 180 – 40

= 140^{0}.

Therefore, no. of sides of the polygon is 9, the exterior angle is 40^{0} and the interior angle is 140^{0}.

**x. **The measure of the exterior angle of a hexagon is (3x-2),(5x+4),(10x+2),(8x+3),(13x-30),(15x+5). Find the measure of each angle.

**Solution:**

Given Exterior angles are (3x-2),(5x+4),(10x+2),(8x+3),(13x-30),(15x+5).

The sum of the exterior angles of any polygon is always equal to 360.

(3x-2)+(5x+4)+(10x+2)+(8x+3)+(13x-30)+(15x+5)=360

54x-18=360

54x=378

x=378/54=7.

Each exterior angle

(3x-2)=(3(7)-2)=21-2=19

(5x+4)=(5(7) +4)=35+4=39

(10x+2)=(10(7) + 2)=70+2=72

(8x+3)=(8(7) + 3)=56+3=59

(13x-30)=(13(7) – 30)=91-30=61

(15x+5)=(15(7) + 5)=105+5=110

Therefore, The measure of exterior angles are 19^{0},39^{0},72^{0},59^{0},61^{0},110^{0}.

**XI. **What is the maximum exterior angle possible for a regular polygon?

**Solution:**

The exterior angle will be maximum when the interior angle is minimal.

Let us consider the interior angle to be 60° since an equilateral triangle is a regular polygon having maximum exterior angle because it consists of the least number of sides.

Exterior angle = 180° – 60°= 120°

Hence, the maximum exterior angle possible for a regular polygon is 120°.