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 area of complex plane figures 2d and 3d figures

A polygon has 44 diagonals, The number of its sides is

  1. 11

  2. 10

  3. 8

  4. 7
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Number of diagonals in a polygon $=\cfrac{n(n-3)}2$
$\implies 44=\cfrac{n(n-3)}2$
$\implies n^2-3n-88=0$
$\implies (n-11)(n-8)=0$
$\implies n=11$ or $n=-8$
Therefore, number of sides in a polygon $=11.$
Hence, A is the correct option.
Multiple choice mathematics and statistics angle and its measurement directed angles

The measure of exterior angle is $40^o$. Find number of side.

  1. $7$
  2. $8$
  3. $9$
  4. $10$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sum of exterior angles of any convex polygon is 360 degrees. For a regular polygon, each exterior angle is 360/n. Here, 360/n = 40, so n = 360/40 = 9.

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon constructions related to a circle construction of polygons construction of tangent to a circle construction of tangents construction of line segment and circle of given radius construction related to lines

The measure of maximum possible exterior angle in a regular polygon is 

  1. $70^o$
  2. $60^o$
  3. $90^o$
  4. $120^o$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Exterior angle of regular polygon = 180-interior angle
exterior angle is maximum when interior angle is minimum.
And we have minimum interior angle for regular triangle that is 60 degree..
So maximum exterior angle will be 180-60=120
So correct answer is Option D
Multiple choice statistics information processing fundamental principle of addition fundamental principles of counting principles of counting

The number of rectangles that can be obtained by joining four of the twelve vertices of a $12$ sided regular polygon is

  1. $66$
  2. $30$
  3. $24$
  4. $15$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
The first vertex can be choosed in $12$ ways and diagonally opposite to it is $1$ vertex. Now for $3rd$ vertex we have $10$ choices and for $4th$ $1.$
However, each rectangle is counted $8$ times.
$\therefore$ No. of ways $=\dfrac{12\times1\times10\times1}{8}$  $=15$ ways.
Hence, the answer is $15.$
Multiple choice maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

Can we construct a rhombus $ABCD$ with $AB=4\ cm$? Its diagonal intersect at the point $O$ and $\angle OAB = 60^0$.

  1. Yes

  2. No

  3. Sometimes yes

  4. Can't say
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given : $AB=4$cm

Diagonal intersect at $O$ and $\angle OAB=60^{o}$ ....... $(1)$
Draw side $AB$ of $4$cm.
In a rhombus, all sides are equal and diagonals bisect the opposite angles
From $(1)$ we get, $\angle A=120^{o}$
$\implies \angle B=60^{o}$ ........... (Adjacent angles are supplementary)
Draw a side $AD$ from A of $4$cm such that $\angle BAD=120^{o}$
Now, from $D$, draw side $DC = 4$cm such that $\angle ADC=60^{o}$
And then join $B-C$ such that $BC=4$cm and $\angle DCB=120^{o}$.
At last we get a rhombus $ABCD$ with length of each side is $4$ cm and diagonals $AC$ and $BD$.
Hence, we can construct a rhombus with $AB=4\ cm$.

Multiple choice maths construction circumscribing and inscribing a circle on a regular hexagon construction of tangent to a circle construction of tangents

Inscribe a regular pentagon in a circle of radius $3\ cm$. The interior angles of the pentagon are:

  1. $54^\circ$
  2. $60^\circ$
  3. $162^\circ$
  4. $108^\circ$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We know that internal angle of regular pentagon is $\cfrac{(n-2)}{n}180^{\circ}$ where n = number of sides.

Here, n = 5.
So, interior angle is $\cfrac{(5-2)}{5}180^{\circ} = 108^{\circ}$

So correct answer is option D

Multiple choice exponent of a prime in n! factorial notation combinatorics and mathematical induction permutations and combinations maths

Let ${T _n}$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If ${T _{n + 1}} - {T _n} = 10$. then the value of $n$ is 

  1. $7$
  2. $5$
  3. $10$
  4. $8$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $T _n=\ ^nC _3$

$T _{n+1}=\ ^{n+1}C _3$
therefore,
$T _{n+1}-T _n=\ ^{n+1}C _3-\ ^nC _3=10$
$\Rightarrow\ ^nC _2+\ ^nC _3-\ ^nC _3=10$        $[\because\ ^nC _r+\ ^nC _{r-1}=\ ^{n+1}C _r]$
$\Rightarrow\ ^nC _2=10$
$\Rightarrow\ ^nC _2=\ ^5C _2$
$\Rightarrow n=5$

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.