Tag: sum of exterior angles of polygons

Questions Related to sum of exterior angles of polygons

The interior angle of a regular polygon is double the exterior angle. Then the number in the polygon is 

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. $6$

  2. $8$

  3. $9$

  4. None of these

Show answer
Correct Option: A
Explanation:

Let the number of sides in the polygon is $n$

Let the measure of exterior angles be $x$ respectively.
$\therefore$   Measure of interior angle $=2x$
So,
$\Rightarrow$  $n\times 2x=(2n-4)\times 90^o$
$\Rightarrow$  $nx=(n-2)\times 90^o$               ----- ( 1 )
Again we know that,
$\Rightarrow$  $nx=360^o$
$\Rightarrow$  $(n-2)\times 90^o=360^o$               [ From ( 1 ) ]
$\Rightarrow$  $n-2=4$
$\Rightarrow$  $n=6$
$\therefore$   The number of sides in the polygon are $6.$

State true or false:
Is it possible to have a regular polygon whose each exterior angle is $\displaystyle 32^{\circ}$
maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. True

  2. False

Show answer
Correct Option: B
Explanation:
Each exterior angle of regular polygon of n sides is given by $\dfrac{360}{n}$ degree
$\text{According to the question}$
$\dfrac{360}{n} degree=32^0$
$\Rightarrow n=\dfrac{360}{32}$
$\Rightarrow n=\dfrac{45}4$
$\text{Which is not possible because number of sides can never be in fraction.}$

State true or false:
Is it possible to have a regular polygon whose each exterior angle is $\displaystyle 20^{\circ}$

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. True

  2. False

Show answer
Correct Option: A
Explanation:
$\text{Each exterior angle of regular polygon of n sides is given by}\dfrac{360}{n} degree$
$\text{According to question}$
$\dfrac{360}{n} degree=20^0$
$\Rightarrow n=\dfrac{360}{20}$
$\Rightarrow n=18$
$\text{Clearly there is a polygon of sides 18 whose each exterior angle is 20}^0$

Two alternate sides of a regular polygon, when produced, meet at a right angle, then find the value of each exterior angle of the polygon.

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. $\displaystyle 45^{\circ}$

  2. $\displaystyle 32^{\circ}$

  3. $\displaystyle 62^{\circ}$

  4. $\displaystyle 15^{\circ}$

Show answer
Correct Option: A
Explanation:

$We\quad know\quad External\quad angles\quad of\quad a\quad regular\quad polygon\quad are\quad equal.$


$\ When\quad the\quad two\quad alternate\quad sides\quad of\quad a\quad regular\quad polygon\quad are\quad produced,\quad they\quad meet\quad at\quad right\quad angle.$

$\ These\quad two\quad extended\quad sides\quad form\quad a\quad triangle\quad with\quad the\quad side\quad of\quad the\quad polygon\quad in\quad between.$

$\ Sum\quad of\quad all\quad interior\quad angles\quad of\quad a\quad \triangle ={ 180 }^{ o }$

$\ \Rightarrow 2\times External\quad angle\quad +\quad { 90 }^{ o }\quad =180$

$\ \Rightarrow 2\times External\quad angle=180-90=90$

$\ \Rightarrow External\quad angle=\dfrac { 90 }{ 2 }$

$ \ \Rightarrow External\quad angle={ 45 }^{ o }
$

State true or false: 

Is it possible to have a regular polygon whose each exterior angle is 40% of a right angle.

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Given, regular polygon whose each exterior angle is 40% of a right angle = $ 90^o \times \dfrac{40}{100} = 36^o $
Sum of all exterior angle of any polygon is $ 360^o $
Now,
$ 36^o \times$  number    of   angles  = $ 360^o $
The number  of  angles =$ 10 $
Any polygon have equal number of angles and sides.
Therefore the number of side of the polygon is 10.
Since the number of sides is an integer, therefore their exist a polygon whose each exterior angle is 40% of a right angle. 

State true or false.
Is it possible to have a regular polygon whose each exterior angle is $32^{\circ}$

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Given, a regular polygon whose each exterior angle is $32^o $
Each exterior angle of a regular polygon = $ \dfrac {360^o}{n} $, where n = number of side
Now,
$ \dfrac {360^o}{n} = 32^o $
$=> n = \dfrac{45}{4} $
Since, n should be an integer, so it is not possible a regular polygon whose each exterior angle is $32^o $

State true or false: 
Is it possible to have a regular polygon whose each exterior angle is $80^o$
80∘

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

Let the number of sides of the polygon is n. (which must be an integer)
If each exterior angle is $ 80^o $, then sum of all exterior angle is $ n \times 80^o $.
And Sum of all exterior angles = $ 180^o $
$=>  n \times 80^o = 180^o $
$=> n = 1.25 $
So, it is not possible to have a regular polygon whose each exterior angle is $ 80^o $

State true or false.
Is it possible to have a regular polygon whose each exterior angle is $20^{\circ}$

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

Given, a regular polygon whose each exterior angle is $20^o $
Each exterior angle of a regular polygon = $ \dfrac {360^o}{n} $, where n = number of side
Now,
$ \dfrac {360^o}{n} = 20^o $
$=> n = 18 $
Since, n should be an integer, so their exist a regular polygon whose each exterior angle is $20^o $

How many sides does a regular polygon have if the measure of an exterior angle is $24^o$?

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. $15$

  2. $12$

  3. $14$

  4. $16$

Show answer
Correct Option: A

Find the measure of exterior angle of a regular polygon of 15 sides

maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

  1. $36^o$

  2. $24^o$

  3. $48^o$

  4. none of the above

Show answer
Correct Option: B
Explanation:

The sum of the exterior angles of a regular polygon is $360^o$

Number of sides of polygon $=15$
As each of the exterior angles are equal,

Exterior angle $=\dfrac{360^o}{15}=24^o$