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 interior angle of a regular polygon exceeds the exterior angle by $ \displaystyle 132^{\circ}  $, then the number of sides of the polygon is :

  1. $15$
  2. $14$
  3. $13$
  4. $12$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the number of sides in the regular polygon be $n$

Thus each interior angle $=$ $\dfrac{(2n-4)\times 90^{\circ}}{n}$
And each exterior angle $=\dfrac{360^{\circ}}{n}$
Lets go according to question:
Therefore, $  \dfrac{(2n-4)\times 90^{\circ}}{n}-\dfrac{360^{\circ}}{n}=132^{\circ}$
$\Rightarrow 180n-360-360=132n$
$\Rightarrow 48n=720$
$\Rightarrow n=\dfrac{720}{48}=15$

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

Let the  formula relation the exterior angle and number of sides of a polygon be given as $nA = 360$.
The measure $A$, in degrees, of an exterior angle of a regular polygon is related to the number of sides, $n$, of the polygon by the formula above. If the measure of an exterior angle of a regular polygon is greater than $50$, what is the greatest number of sides it can have?

  1. 5

  2. 6

  3. 7

  4. 8
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Sum of exterior angles for any polynomial is always $360$. 

Since polynomial has $n$ angles, each with exterior angle is $A$, then 
sum of exterior angles will be $nA$ 
Given, $nA = 360$ 
$\therefore A=\dfrac { 360 }{ n }$  
We are given that: $A > 50$ 
$\Rightarrow \dfrac { 360 }{ n } >50$ 
$\Rightarrow 360 > 50n$ 
$\Rightarrow n<\dfrac { 360 }{ 50 }$  
$\Rightarrow n < 7.2$ 
Hence, the greatest number of angles polygon can have is $7$.

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

Which one of the following statements is not correct?

  1. if the exterior angle of a regular polygon is $30$ it has $12$ sides

  2. if the interior and exterior angles of a regular polygon are all equal, it is a rectangle

  3. if the exterior angle of a regular polygon is greater than its interior angle, it is an equilateral triangle

  4. in a regular pentagon, the exterior angle is half of the interior angle
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

A) Exterior angle of m-gon$=\cfrac { (m180)-(m-2)180 }{ m } $

If $m=12$, 
$\Longrightarrow $ Exterior angle$=30$.
Therefore A is correct.

B) If ABCD is a rectangle,
Interior$=$Exterior angle$={ 90 }^{ 0 }$ .
Therefore B is correct.

C) In equilateral triangle exterior angle ($120$)$>$ interior angle$60$.
Whereas in others it is less than or equal to interior angle.
Therefore C is true.

D) Exterior angle of pentagon$=72$.
Interior angle of pentagon$=108$.
Exterior angle $\neq \cfrac { 1 }{ 2 } $interior angle.
Therefore D is incorrect.

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

The sum of the exterior angles of a hexagon is?

  1. $360^{\circ}$
  2. $540^{\circ}$
  3. $720^{\circ}$

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

Number of sides in hexagon $=6$
Sum of the interior angles of a polygon$=(n-2)\pi$
$n(Interior\ Angle)=(n-2)\pi$
$\Rightarrow $ Interior Angle $= \dfrac{4}{6}\pi$

Interior Angle $= 120^\circ$
Exterior Angle $=180- $Interior Angle
$\Rightarrow$ Exterior angle $=60^\circ$
Sum of Exterior angle $=6 \times$ Exterior Angle $=360^\circ$

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

How many sides does a regular polygon have if the measure of an exterior angle is $24^{0}$?

  1. $14$
  2. $13$
  3. $15$
  4. $18$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Here, let the number of sides of the polygon be $n$ 

So, the number of exterior angles is $n$ 
Since it is a regular polygon, each exterior angles are equal to one another.
$ \therefore$ The sum of the exterior angles $={ 360 }^{ o }$
$ \therefore$  Each angle $=\theta =\dfrac { { 360 }^{ o } }{ n }$ 
$\Longrightarrow n=\dfrac { { 360 }^{ o } }{ \theta  } $ 
Here $\theta ={ 24 }^{ o } $ 
$ \therefore  n=\dfrac { { 360 }^{ o } }{ { 24 }^{ o } } =15$
Hence, the answer is $15$.

Multiple choice maths arithmetic progressions complete the a.p series with given information properties of an ap problems on ap

${ T } _{ m }$ denotes the number of triangles that can be formed with the vertices of a regular polygon of m sides. If ${ { T } _{ m+1 } }-{ { T } _{ m } }=15,$ then $m=$

  1. $3$
  2. $6$
  3. $9$
  4. $12$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

T_m = mC3. The condition T_{m+1} - T_m = 15 becomes (m+1)C3 - mC3 = 15, which simplifies to mC2 = 15. Solving m(m-1)/2 = 15 gives m^2 - m - 30 = 0, so (m-6)(m+5)=0. Thus m=6.

Multiple choice maths construction of quadrilaterals trapeziums and kites quadrilaterals and their properties closed figures

The ratio of the measures of the consecutive angles of a quadrilateral is 1:2:3:4. What type of quadrilateral is it

  1. Trapezium

  2. Kite

  3. Parallelogram

  4. Rectangle
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of angles in a quadrilateral is 360 degrees. If the ratio is 1:2:3:4, the angles are 36, 72, 108, and 144 degrees. Since one pair of consecutive angles (72+108) sums to 180, it is a trapezium.

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

If the angles A, B, C and D of a quadrilateral ABCD in the same order are in the ratio 3 : 7 : 6 : 4, then ABCD is a 

  1. parallelogram

  2. rhombus

  3. trapezium

  4. kite
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let angles be $3x, 7x, 6x$ and $4x$
Total sum of angles of a quadrilateral = $360^{\circ}$
$\Rightarrow 3x + 7x + 6x + 4x = 360^{\circ}$
$\Rightarrow 20x = 360^{\circ}$
$\Rightarrow x = 18^{\circ}$
$\therefore$ Angles are $ 3\, \times\, 18^{\circ} = 54^{\circ}$
 $ 7\, \times\, 18^{\circ} = 126^{\circ}$
 $ 6\, \times\, 18^{\circ} = 108^{\circ}$
 $ 4\, \times\, 18^{\circ} = 72^{\circ}$
All the angles of the figure ABCD are different, thus it is a trapezium.
Hence, option 'C' is correct.

Multiple choice maths when lines join trapeziums and kites quadrilaterals and their properties closed figures

If the angles $A, B, C, D$ of a quadrilateral , taken in order are in the ratio $7:13:12:8$, then $ABCD$ is:

  1. rhombus

  2. parallelogram

  3. trapezium

  4. kite
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let the angles be $7x, 13x, 12x$ and $8x$
Then, $7x+13x+12x+8x={360}^{o}$
$\Rightarrow$ $40x={360}^{o}$ $\Rightarrow$ $x={9}^{o}$
$\therefore$ $40x={360}^{o}$
$\therefore$ The angles taken in order are ${63}^{o}, {117}^{o}, {108}^{o}, {72}^{o}$ 
This shows that tow pairs of adjacent angles are supplementary $({63}^{o}+{117}^{o}={108}^{o}$ and ${108}^{o}+{72}^{o}={180}^{o}$), but opposite angles are not equal.
Therefore, the given quadrilateral will be a trapezium.

Multiple choice

What is the name of the theorem that states that the sum of the interior angles of a polygon with n sides is (n-2) * 180 degrees?

  1. Brahmagupta's Theorem

  2. Bhaskara's Theorem

  3. Aryabhata's Theorem

  4. Varahamihira's Theorem
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Brahmagupta's Theorem states that the sum of the interior angles of a polygon with n sides is (n-2) * 180 degrees. This theorem is also known as the polygon angle sum formula.

Multiple choice

What is the name of the theorem that states that the sum of the exterior angles of a polygon is equal to 360 degrees?

  1. Brahmagupta's Theorem

  2. Pythagorean Theorem

  3. Euler's Formula

  4. Descartes' Theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Descartes' Theorem states that the sum of the exterior angles of a polygon is equal to 360 degrees.

Multiple choice

What is the name of the theorem that states that the sum of the exterior angles of a polygon is equal to 360 degrees?

  1. Brahmagupta's Theorem

  2. Pythagorean Theorem

  3. Euler's Formula

  4. Descartes' Theorem
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Descartes' Theorem states that the sum of the exterior angles of a polygon is equal to 360 degrees.

Multiple choice

The sum of the interior angles of a quadrilateral is always:

  1. 360 degrees

  2. 540 degrees

  3. 720 degrees
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of the interior angles of a quadrilateral is always 360 degrees.