Unit 2MM1G3 a

Sum of the Interior and Exterior angles in Polygons!

Interior Angles of a Polygon

http://mathopenref.com/polygoninteriorangles.html

Exterior Angles of a Polygon

http://mathopenref.com/polygonexteriorangles.html

Interior/Exterior Angle Relationship

http://mathopenref.com/polygonanglerelation.html

Since the sum of the measures of the interior angles of a triangle is 180o , we can use this fact to help us find the sum of the measures of the interior angles of any convex n-gon.

Example 1: Find the sum of the measures of the interior angles of quadrilateral ABCD below.

Begin by drawing diagonal AC.

As you can see, this diagonal divides quadrilateral ABCD into two triangles. Therefore, the sum of the measures of the interior angles is

180o x 2 triangles = 360o .

BA

CD

AB

CD

Example 2: Find the sum of the measures of the interior angles of a pentagon.

E D

C

B

A

Draw diagonals AC and AD.

These diagonals divide the pentagon into three triangles. Therefore, the sum of the measures of the interior angles of the pentagon is

180o x 3 triangles = 540o

The same method can be applied to convex polygons with many sides.

Polygon Number of sides

Number of triangles

Sum of the interior angles

Triangle 3 1 1 x 180o = 180o

Quadrilateral 4 2 2 x 180o = 360o

Pentagon 5 3 3 x 180o = 540o

Hexagon 6 4 4 x 180o = 720o

Heptagon 7 5 5 x 180o = 900o

Octagon 8 6 6 x 180o = 1080o

n-gon n n – 2 (n – 2) x 180o

Therefore, the sum of the measures of the interior angles of any convex polygon can be found by using (n – 2) x 180o where n is the number of sides.

Example 3: Find the sum of the measures of the interior angles of a decagon.

Solution: A decagon has 10 sides. Using the formula (n – 2) x 180o, we can find the sum.

(10 – 2) x 180o = 8 x 180o = 1440o

Therefore, the sum of the measures of the interior angles of a decagon is 1440o .

Example 4: Find the value of x in the following figure.

114°

x°

135° 102°

85°

115°

Solution: Since the figure has six sides, the sum of the measures of the interior angles is 720o .

(n – 2) · 180° = (6 – 2) · 180° = 720°

Solving for x:135° + 102° + 85° + 115° + 114° + x° = 720°

551° + x° = 720° x° = 169°

114°

x°

135° 102°

85°

115°

If we know the sum of the measures of the interior angles of a polygon, we can work backwards to find how many sides it has.

Example 5: The sum of the measures of the interior angles of a convex polygon is 720º. How many sides does it have?

Begin with the formula

Now, solve for n (the number of sides).

Therefore, the polygon has 6 sides.

€

n − 2( )180o = 720o

€

n − 2( )180o

180o =720o

180o

n − 2 = 4

n = 6

A regular polygon is both equilateral and equiangular. The measure of one interior angle can be found by dividing the sum of the measures of the interior angles by the number of sides.

Regular polygon

Number of sides

Number of triangles

Sum of the interior angles

Measure of one interior angle

Triangle 3 1 1 x 180o = 180o 180o / 3 = 60o

Quadrilateral 4 2 2 x 180o = 360o 360o / 4 = 90o

Pentagon 5 3 3 x 180o = 540o 540o / 5 = 108o

Hexagon 6 4 4 x 180o = 720o 720o / 6 = 120o

Heptagon 7 5 5 x 180o = 900o 900o / 7 = 128.57o

Octagon 8 6 6 x 180o = 1080o 1080o / 8 = 135o

n-gon n n – 2 (n – 2) x 180o n

n 01802

Example 6: Find the measure of one interior angle in a regular octagon.

Solution: An octagon has 8 sides. The sum of the measures of the interior angles is

(8 – 2) x 180o = 1080o

To find the measure of one interior angle, divide this sum by the number of angles.

Therefore, each interior angle in a regular octagon has a measure of 135o .

€

(8 − 2) ×180o

8=

1080o

8=135o

If we know the measure of one interior angle of a regular convex polygon, then we can work backwards to find how many sides it has.

Example 7: The measure of one interior angle of a regular convex polygon is 144º. How many sides does it have?

Begin with the formula for the measure of one interior angle.

Now solve for n (the number of sides)

Therefore, the polygon has 10 sides.

€

n − 2( )180o

n=144o

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n − 2( )180o =144on

180on − 360o =144on

36on = 360o

n =10

Summary

The sum of the measures of the interior angles of a convex n-gon = (n – 2) x 180o .

The measure of one interior angle of a regular convex n-gon =

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n − 2( )180o

n

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Exterior Angles of a Polygon

Examples

An exterior angle of a polygon is formed by extending one side of the polygon.

An exterior angle and its adjacent interior angle are supplementary.

Interiorangle

exteriorangle

1 2

ary.supplement are 2 and 1

1

3

4

2 5

6

Example 1: Find the sum of the measures of the exterior angles of the triangle below.

0180321 know that weFirst, mmm

ary.supplement are angles

interior adjacent andexterior theknow that also We

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So, m∠1+m∠4 =180o

m∠2 +m∠5 =180o

m∠3 +m∠6 =180o

We can add the equations together.

Since we know the sum of the interior angles of the triangle is 180°, we can substitute and solve.

So, the sum of the measures of the exterior angles of the triangle is 360°.

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m∠1+m∠4 =180o

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m∠2 +m∠5 =180o

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m∠1+m∠2 +m∠3+m∠4 +m∠5 +m∠6 = 540o

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m∠3+m∠6 =180o

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180o +m∠4 +m∠5 +m∠6 = 540o

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m∠4 +m∠5 +m∠6 = 360o

Example 2: Find the sum of the measures of the exterior angles of a quadrilateral.

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Remember, the sum of the measures of the interior

angles of a quadrilateral is 4 − 2( )180o = 360o.

8

7

6

5

4

3 2

1

€

So, m∠1+m∠2 +m∠3 +m∠4 = 360o

Again, we know that each exterior angle and its adjacent interior angle are supplementary.

1

2

5

4

3

8

7

6

0

0

0

0

0

72087654321

18084

18073

18062

18051

mmmmmmmm

mm

mm

mm

mm

0

00

0

3608765

7208765360

solve. and substitute can we

,3604321know weSince

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mmmm

1

23

4

5

6

7

8

Notice that the sum of the measures of the exterior angles of the quadrilateral is also 360º.

Further exploration shows us that the sum of the measures of the exterior angles of any convex polygon is always 360º.

Example 3:

In a pentagon, there are 5 exterior/interior angle pairs. Each pair is supplementary.

5 x 180° = 900°

We know the sum of the measures of the interior angles is (5 – 2) 180° or 540°.

900° - 540° = 360° (the sum of the measuresof the exterior angles.

Interiorangle

Interiorangle

Interiorangle

Interiorangle

Interiorangle

exteriorangle

exteriorangle

exteriorangle

exteriorangle

exteriorangle

We can also easily find the measure of one exterior angle of a regular convex polygon.

Example 4: Find the measure of one exterior angle of a regular pentagon.

Solution:

Since the sum of the measures of the

exterior angles of ANY convex polygon

is 360º, then we simply divide by the

number of sides.

Therefore, the measure of each exterior angle of a regular pentagon is 72º

00

725

360

exteriorangle

exteriorangle

exteriorangle

exteriorangle

exteriorangle

In a regular polygon, we can use the formula where n is the number of sides to find the measure of each exterior angle.

Example 4: Find the measure of one exterior angle of a regular convex 15-gon.

Solution: The sum of the measures of the exterior angles of a regular 15-gon is 360°.

Therefore, the measure of each exterior angle of a regularconvex 15-gon is 24°.

€

360o

15= 24o

€

360o

n

Example 5: Each exterior angle of a certain regular convex polygon measures 20º. How many sides does the polygon have?

Solution: We can work the formula backwards to find the number of sides.

Therefore, the regular polygon has 18 sides.

€

360o

n

€

360o

n= 20o

n =18

QUIZ

B

D

A

C

D

C

C

B

Calculate your score!!