Mathematics

Polygons and Angles

100 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 polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

If the difference between an interior angle of a regular polygon of $\displaystyle \left ( n+1 \right )$ sides and an interior angle of a regular polygon of $n$ sides is $\displaystyle 4^{\circ}$; find the value of $n$. Also, state the difference between their exterior angles.

  1. $\displaystyle n =9$ and difference between exterior angles $\displaystyle 4^{\circ}$
  2. $\displaystyle n =5$ and difference between exterior angles $\displaystyle 22^{\circ}$
  3. $\displaystyle n =11$ and difference between exterior angles $\displaystyle 12^{\circ}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
An interior angle of (n + 1) sided regular polygon = $ \dfrac{180^o((n+1) -2)}{(n+1)} $
An interior angle of n sided regular polygon = $ \dfrac{180^o(n-2)}{n} $
Their difference is $ 4^o $
So, $\dfrac{180^o((n+1) -2)}{(n+1)} - \dfrac{180^o(n-2)}{n}= 4^o$
$=> 45 [  \dfrac{(n-1)}{(n+1)} -  \dfrac{(n-2)}{n} ]= 1 $ 
$=> 45 \dfrac{2}{n(n+1)} = 1 $
$=> n^2 + n -90 = 0$
$=> (n-9)(n+10) = 0$
$=> n = 9, -10$ 
Since n should be a positive number. So, $n = 9$
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 $\displaystyle \frac{1}{8}$ of a right angle.

  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 $ \dfrac{1}{8}$ of a right angle = $ \dfrac {1}{8} \times 90^o = \dfrac {45^o}{4} $
Each exterior angle of a regular polygon = $ \dfrac {360^o}{n} $, where n = number of side
Now,
$ \dfrac {360^o}{n} = \dfrac {45^o}{4}  $
$=> n = 8 $
Since, n should be an integer, so their exist a regular polygon whose each exterior angle is $ \frac{1}{8}$ of a right angle.

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

Three of the exterior angles of a hexagon are $40^{\circ}$, $51^{\circ}$ and $86^{\circ}$. If each of the remaining exterior angles is $x^{\circ}$, find the value of $x$.

  1. $58$
  2. $61$
  3. $65$

  4. none of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Three of the exterior angles of a hexagon are $ 40^o, 51^o$  and  $86^o $. Each of the remaining exterior angles is $ x^o $.
Sum of all exterior angle of any polygon is $ 360^o $
$ 40^o + 51^o + 86^o + 3 \times x^o = 360^o $
$ => 3 \times x^o = 183^o $
$ => x^o = 61^o $

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

The sides of a hexagon are produced in order. If the measures of exterior angles so obtained are $\displaystyle (6x-1)^{\circ}, (10x+2)^{\circ}, (8x+2)^{\circ}, (9x-3)^{\circ}, (5x+4)^{\circ}$ and $(12x+6)^{\circ};$. Find each exterior angle.

  1. $41^{\circ}, 62^{\circ}, 58^{\circ}, 60^{\circ}, 39^{\circ} , 90^{\circ}$
  2. $41^{\circ}, 86^{\circ}, 56^{\circ}, 60^{\circ}, 39^{\circ} , 80^{\circ}$
  3. $41^{\circ}, 72^{\circ}, 58^{\circ}, 60^{\circ}, 39^{\circ} , 90^{\circ}$
  4. $41^{\circ}, 82^{\circ}, 60^{\circ}, 60^{\circ}, 36^{\circ} , 100^{\circ}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sum of the exterior angles of any polygon is always equal to 360.
Exterior angles are $\displaystyle (6x-1)^{\circ}, (10x+2)^{\circ}, (8x+2)^{\circ}, (9x-3)^{\circ}, (5x+4)^{\circ}  and  (12x+6)^{\circ} $
Now, 
 $\displaystyle (6x-1)^{\circ}+ (10x+2)^{\circ}+ (8x+2)^{\circ} + (9x-3)^{\circ} + (5x+4)^{\circ} +  (12x+6)^{\circ} = 360^o $
$ => (50x + 10)^o = 360^o $
$ => x = 7 $
Each Exterior angle 
$ => (6x -1)^o = 6 \times 7 -1 =41^o $
$ => (10x +2)^o = 10 \times 7 +2 =72^o $
$ => (8x +2)^o = 8 \times 7 +2 =58^o $
$ => (9x -3)^o = 9 \times 7 -3 =60^o $
$ => (5x +4)^o = 5 \times 7 +4  =39^o $
$ => (12x +6)^o = 12 \times 7 + 6 =90^o $

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. Find the number of sides of the polygon. 

  1. $3$
  2. $8$
  3. $2$
  4. $9$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In a regular polygon all the exterior angles have the same measure. 


When two alternate sides of a polygon are extended a triangle.

If AB, BC and CD are the sides of a regular polygon and AB and CD when produced meet at P forming a right triangle.

Now, in $ \triangle CPB, \angle PCB = \angle PBC = 45^o $

Therefore, exterior angle of the polygon = $ 45^o $
Exterior angle of a regular polygon = $ \dfrac {360^o}{n} $
$=> 45^o = \dfrac {360^o}{n} $
$ => n = 8 $ 
Number of sides of the polygon = $8$

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 interior angle is $\displaystyle 175^{\circ}$

  1. True

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

Each interior angle of regular polygon of $n$ sides is given by $\dfrac{180^o(n-2)}{n}$
According to question

$\dfrac{180^o(n-2)}{n} =175^0$
$\Rightarrow 180^o(n-2)=175n$
$\Rightarrow 180n-360^o=175n$
$\Rightarrow 5n=360^o$
$\Rightarrow n=72$
Clearly there is a polygon of sides $72$ whose each interior angle is $175^0$

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

The sum of the interior angles of a polygon is four times the sum of its exterior angles. Find the number of sides in the polygon.

  1. $10$
  2. $12$
  3. $8$
  4. $7$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of the interior angles of a polygon is four times the sum of its exterior angles.
The sum of the exterior angles of a polygon is always equal to $360^o$.
The sum of the interior angles of polygon = $180 (n-2)$
=> $180 (n-2) = 4 \times 360$
=> $n -2 = 8$ 
=> $n =10$ 
Number of sides in the polygon = $10$

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

There is a regular polygon whose each interior angle is $175^{\circ}$

State true or false.

  1. True

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

Given, a polygon whose each interior angles is $ 175^o $
Sum of interior angles of a polygon is =  $ 180^o (n-2) $
Each interior angle of a polygon = $ \dfrac {180^o (n-2)}{n} $
$ \dfrac {180^o (n-2)}{n}  = 175^o $
$  180^o n - 175^o n = 360^o $
$ n = \dfrac {360}{5} $
$ n = 72 $
Since, n (number of sides) is an integer, therefore there exist a polygon whose each interior angles is $ 175^o $

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

Find the sum of exterior angles obtained on producing, in order, the sides of a polygon with 7 sides.

  1. $360^{\circ}$
  2. $340^{\circ}$
  3. $380^{\circ}$
  4. $390^{\circ}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

No matter what type of polygon, the sum of the exterior angles is always equal to $360^o$.
It does not depends upon number of sides of polygon.  

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

How many sides does a polygon have if the sum of the measures of its internal angles is five times as large as the sum of the measures of its exterior angles?

  1. $20$
  2. $12$
  3. $15$
  4. $10$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Sum of measure of Interior angles of a regular polygon is calculated as,
$(n-2)180$ where,
n: Number of sides of a regular polygon.
Sum of exterior angles of a regular polygon always add up to $360^{o}$
$\therefore$ ,$(n-2)180=5(360)$
$\therefore n=12$

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

Two times the interior angle of a regular polygon is equal to seven times is exterior angle. Find the interior angle of the polygon and the number of sides in it.

  1. $130^{\circ}$ and n $=$ 9
  2. $140^{\circ}$ and n $=$ 9
  3. $160^{\circ}$ and n $=$ 9
  4. $170^{\circ}$ and n $=$ 9
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Two times the interior angle of a regular polygon is equal to seven times is exterior angle.
Each Interior angle of a polygon = $ \dfrac {180^o (n-2)}{n} $
Each Exterior angle of a polygon = $ \dfrac{360^o}{n} $
Now,
$ 2 \times \dfrac {180^o (n-2)}{n} = 7 \times  \dfrac{360^o}{n}  $
$=> n -2 = 7 $
$=> n = 9 $
Number of sides of polygon is 9.
Each Interior angle of a polygon = $ \dfrac {180^o (n-2)}{n} = \dfrac {180^o (9-2)}{9} = 140^o  $

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

The measurement of each angle of a polygon is $160$$^o$. The number of its sides is ?

  1. $15$
  2. $18$
  3. $20$
  4. $30$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given, measure of each angle of a polygon $=160^o$
Exterior angle $= 180^o -$ Interior angle
$= 180^o - 160^o = 20^o$
$\therefore$ Number of sides $= \displaystyle \frac{360^o}{\text{Exterior angle}} = \frac{360}{20} = 18$
Therefore, number of sides of polygon are $18$.
Multiple choice maths polygons exterior angles of polygon sum of exterior angles of polygons exterior angles of a polygon

The ratio of the measure of an exterior angle of a regular $7:2$ nonagon to the measure of one of its interior angles is:

  1. $7:2$
  2. $2:7$
  3. $4:3$
  4. $3:4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $7a$ be the interior angle

and $2a$ be the exterior angle
Therefore, $ 7a+2a=180^{0}$
$\Rightarrow 9a=180^{0}$
$\Rightarrow a=20^{0}$
So, $2a=2\times 20$
$=40^{0}$
and $7a=7\times 20$
$=140^{0}$
For a regular polygon of $n$ sides, each exterior angle has a measure of $\dfrac{360}{n}$ degrees.

The measure of each interior angle is $140^{0}$.
Since the exterior angle of each angle has measure $40^{0}$, then the number of sides $n$.
$=\dfrac{360}{n}$
$=9$ sides.

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

A regular polygon is inscribed in a circle. If a side subtends an angle of $30^{\circ}$ at the centre, what is the number of its sides?

  1. $10$
  2. $8$
  3. $6$
  4. $12$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For a polygon of 'n' sides, the angle subtended at the centre is $ \dfrac {{360}^{o}}{n} $

Given, angle at the centre $ = {30}^{o} $
$ => \dfrac {{360}^{o}}{n}= {30}^{o} $
$ => n = 12 $

Hence, the polygon has $ 12 $ sides.

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

Exterior angles of a regular polygon is one-third of its interior angle. Find number of sides in polygon.

  1. 10

  2. 8

  3. 6

  4. 9
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
If we take $n$ as the number of sides of polygon and $E$ be the exterior angle and $I$ be the interior angle.

$\Rightarrow$   $E+I=180^\circ$  

According to the given question we get,
$\Rightarrow$  $E=\dfrac{1}{3} I$

$\Rightarrow$  So, $I=3E$

$\therefore$  $E+3E=180^\circ$

$\Rightarrow$  $E = 45^\circ$

$\Rightarrow$  Interior angle  $=135^\circ$

$\Rightarrow$  Interior angle $=\dfrac {(n-2)\times 180}{n}$

$\Rightarrow$  $135n=180n-360$

$\Rightarrow$  $-45=-360$

$\Rightarrow$  $n=8$

$\therefore$  Number of sides in polygon are $8$.