Tag: maths

Questions Related to maths

Two alternate sides of a regular polygon, when produced, meet at a right angle. Find the number of sides of the polygon. 

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

  1. $3$

  2. $8$

  3. $2$

  4. $9$

Show answer
Correct Option: B
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 = $8$

State true or false:
Is it possible to have a regular polygon whose each interior angle is $\displaystyle 175^{\circ}$

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

  1. True

  2. False

Show answer
Correct Option: A
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 $175^0$

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.

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

  1. $10$

  2. $12$

  3. $8$

  4. $7$

Show answer
Correct Option: A
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 = $10$

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

State true or false.

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

  1. True

  2. False

Show answer
Correct Option: A
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 $

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

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

  1. $360^{\circ}$

  2. $340^{\circ}$

  3. $380^{\circ}$

  4. $390^{\circ}$

Show answer
Correct Option: A
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.  

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?

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

  1. $20$

  2. $12$

  3. $15$

  4. $10$

Show answer
Correct Option: B
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 $360^{o}$
$\therefore$ ,$(n-2)180=5(360)$
$\therefore n=12$

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.

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

  1. $130^{\circ}$ and n $=$ 9

  2. $140^{\circ}$ and n $=$ 9

  3. $160^{\circ}$ and n $=$ 9

  4. $170^{\circ}$ and n $=$ 9

Show answer
Correct Option: B
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  $

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

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

  1. $15$

  2. $18$

  3. $20$

  4. $30$

Show answer
Correct Option: B
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 $18$.

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:

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

  1. $7:2$

  2. $2:7$

  3. $4:3$

  4. $3:4$

Show answer
Correct Option: B
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.

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?

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

  1. $10$

  2. $8$

  3. $6$

  4. $12$

Show answer
Correct Option: D
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.