Mathematics

Polygons and Angles

102 Questions

Polygons and angles focus on calculating interior and exterior angle sums. Questions also cover properties of regular shapes like hexagons and nonagons. This geometry topic is essential for quantitative aptitude sections in major competitive exams.

Interior angle sumsExterior angle sumsRegular polygonsPolygon propertiesTriangle formation

Polygons and Angles Questions

Multiple choice maths congruence introduction to shapes similarity of triangles introduction to similar triangles

Two polygons of the same number of sides are similar if all the corresponding interior angles are:

  1. Equal

  2. Proportional

  3. Congruent

  4. Cannot say
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Two polygons of the same number of sides are similar, if: 

(a) Their corresponding angles are equal. 
(b) Their corresponding sides are in the same ratio (Proportional).
Hence, nothing can be said about two given polygons when only the angles are congruent, is known.

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

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

  1. $6$
  2. $8$
  3. $9$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $6.$

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Each exterior angle of regular polygon of n sides is given by $\dfrac{360}{n}$ degree
$\text{According to the question}$
$\dfrac{360}{n} degree=32^0$
$\Rightarrow n=\dfrac{360}{32}$
$\Rightarrow n=\dfrac{45}4$
$\text{Which is not possible because number of sides can never be in fraction.}$
Multiple choice maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\text{Each exterior angle of regular polygon of n sides is given by}\dfrac{360}{n} degree$
$\text{According to question}$
$\dfrac{360}{n} degree=20^0$
$\Rightarrow n=\dfrac{360}{20}$
$\Rightarrow n=18$
$\text{Clearly there is a polygon of sides 18 whose each exterior angle is 20}^0$
Multiple choice maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

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.

  1. $\displaystyle 45^{\circ}$
  2. $\displaystyle 32^{\circ}$
  3. $\displaystyle 62^{\circ}$
  4. $\displaystyle 15^{\circ}$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 }
$

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

State true or false: 

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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. 

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

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $

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

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $

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

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

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
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 $

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

An exterior angle of regular polygon is $\displaystyle 12^{\circ}$ the sum of all the interior angles is

  1. $\displaystyle 4040^{\circ}$
  2. $\displaystyle 5040^{\circ}$
  3. $\displaystyle 6040^{\circ}$
  4. $\displaystyle 7040^{\circ}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given the exterior angle of regular polygon is 12

We know each  exterior angle of regular polygon=$\dfrac{360}{n}$ where n is the sides of polygon
$\dfrac{360}{n}=12\Rightarrow n=30$
we know that interior angle of  regular polygon=$180(n-2)=180(30-2)=5040^{0}$

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

The measure of the external angle of a regular hexagon is 

  1. ${\pi/3}$
  2. ${\pi/4}$,
  3. ${\pi/6}$

  4. None
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow$ Sum of exterior angles of a regular hexagon $=360^o$

$\Rightarrow$  Number of sides of regular hexagon $=6$
$\Rightarrow$  The measure of the external angle of a regular hexagon $=\dfrac{360^o}{6}=60^o$
In radian $=60^o\times \dfrac{\pi}{180^o}=\dfrac{\pi}{3}$