FIND THE MEASURE OF EACH INTERIOR ANGLE OF A REGULAR POLYGON

Problem 1 :

For the following regular polygon given below,

find

(i)  Number of sides that the polygon has

(ii) Number of angles

(iii)  Size of each angle.

Solution :

a) Equilateral triangle

Number of sides = 3, number of angles = 3

b) Square

Number of sides = 4, number of angles = 4

c) Pentagon

Number of sides = 5, number of angles = 5

d) Hexagon

Number of sides = 5, number of angles = 5

e) Octagon

Number of sides = 8, number of angles = 8

f) Decagon

Number of sides = 10, number of angles = 10

Problem 2 :

Find the size of each angle of a regular 12 sided polygon.

Solution :

Sum of interior angles of 12 sided polygon = (12 - 2) x 180

= 10 x 180

= 1800

Each angle measure = 1800/12

= 150

So, the size of each angle of a 12 sided regular polygon is 150 degree.

Problem 3 :

Each exterior angle of a regular polygon is 20⁰. Work out the number of sides of the polygon.

Solution :

Exterior angle of a regular polygon = 20

Interior angle = 180 - 20

Each interior angle = 160

So, the polygon will have 18 sides.

Problem 4 :

The trampoline shown is shaped like a regular dodecagon

a. Find the measure of each interior angle.

b. Find the measure of each exterior angle.

Solution :

Number of sides of dodecagon = 12

a) Sum of interior angles of polygon = (n - 2) 180

= (12 - 2) 180

= 10(180)

= 1800

Each interior angle = 1800/number of sides

= 1800/12

= 150

b) Interior angle + exterior angle = 180

150 + exterior angle = 180

exterior angle = 180 - 150

= 30

Problem 5 :

The floor of the gazebo shown is shaped like a regular decagon. Find the measure of each exterior angle of the regular decagon. Find the measure of each exterior angle.

Solution :

Number of sides of decagon = 10

Sum of interior angles of polygon = (n - 2) 180

= (10 - 2) 180

= 8(180)

= 1440

Each interior angle = 1440/number of sides

= 1440/10

= 144

b) Interior angle + exterior angle = 180

144 + exterior angle = 180

exterior angle = 180 - 144

= 36

So, the exterior angle is 36 degree.

Problem 6 :

Write a formula to find the number of sides n in a regular polygon given that the measure of one interior angle is x°.

Solution :

Sum of interior angle + exterior angle = 180

x + exterior angle = 180

One interior angle = (n - 2) (180/n)

x = (n - 2) (180/n)

xn = 180n - 360

360 = 180n - xn

Factoring n, we get

360 = n(180 - x)

n = 360 / (180 - x)

Problem 7 :

Write a formula to find the number of sides n in a regular polygon given that the measure of one exterior angle is x°.

Solution :

Sum of interior angle + exterior angle = 180

One interior angle = (n - 2) (180/n)

(n - 2) (180/n) + x = 180

180 n - 360 + nx = 180n

180n - 180n + nx = 360

nx = 360

n = 360/x

Problem 8 :

Which of the following angle measures are possible interior angle measures of a regular polygon? Explain your reasoning. Select all that apply.

a) 162°     b) 171°    c) 75°     d) 40°

Solution :

One interior angle of polygon = (n - 2) (180/n)

Option a : 

162 = (n - 2) (180/n)

162n = 180(n - 2)

162n = 180n - 360

360 = 180n - 162n

360 = 18n

n = 360/18

n = 20

Option b :

171 = (n - 2) (180/n)

171n = 180(n - 2)

171n = 180n - 360

360 = 180n - 171n

360 = 9n

n = 360/9

n = 40

Option c :

75 = (n - 2) (180/n)

75n = 180(n - 2)

75n = 180n - 360

360 = 180n - 75n

360 = 105n

n = 360/105

n = 3.42

So, 75 cannot be interior angle of polygon.

Option d :

40 = (n - 2) (180/n)

40n = 180(n - 2)

40n = 180n - 360

360 = 180n - 40n

360 = 140n

n = 360/140

n = 2.5

So, 40 cannot be interior angle of polygon.