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

Is it possible to have a regular polygon with measure of each exterior angle as $22^o$?

  1. not possible

  2. possible

  3. cannot be determined

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

Since the number of sides of a regular polygon
$=\dfrac {360}{\text {Exterior angle}}$
$\therefore$ The number of sides of a regular polygon
$=\dfrac {360}{22}[\because$ Exterior angle $=22^o$, given]
$=\dfrac {180}{11}$
Which is not a whole number.
$\therefore$ A regular polygon with measure of each exterior angle as $22^o$ is not possible.

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 octagon is 

  1. ${\pi/4}$
  2. ${\pi/6}$
  3. ${\pi/8}$
  4. ${\pi/12}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\Rightarrow$  The sum of the exterior angles of regular octagon is $360^o$.

$\Rightarrow$ Number of sides of octagon $=8$
$\Rightarrow$  The measure of the external angles $=\dfrac{360^o}{8}=45^o$
In radian $=45^o\times \dfrac{\pi}{180^o}=\dfrac{\pi}{4}$
$\therefore$  The measure of the external angle of a regular octagon is $\dfrac{\pi}{4}$

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

The exterior angle of a regular polygon is one-third of its interior angle. How many sides does the polygon has?

  1. $10$
  2. $8$
  3. $9$
  4. $13$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let no of sides of the polygon is $n$ 

Exterior angle will be $\dfrac{360}{n}$
Interior angle will be $\left ( 180-\dfrac{360}{n}\right)$
Exterior angle is $\dfrac{1}{3}$ of the interior angle
$\Rightarrow \dfrac{360}{n}=\dfrac{1}{3} \left (180-\dfrac{360}{n}\right)$
$\Rightarrow n=8$

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

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 $360^{o}$
$\therefore$ ,$(n-2)180=5(360)$
$\therefore n=12$