Question 1: Calculate the sum of all the interior angles of a polygon having:

i)  6 sides     ii)  8 sides     iii)  14 sides     iv)  20 sides

Answer:

Sides = n Sum of the interior angles =(2n-4)\times 90^{\circ}
i) 6 (2\times 6-4)\times 90^{\circ} =720^{\circ} 
ii) 8 (2\times 8-4)\times 90^{\circ} =1080^{\circ} 
iii) 14 (2\times 14-4)\times 90^{\circ} =2160^{\circ} 
iv) 20 (2\times 20-4)\times 90^{\circ} =3240^{\circ} 

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Question 2: Find the number of sides of a polygon, the sum of whose interior angles is:

i) 540^{\circ}             ii)  1080^{\circ}             iii)  1980^{\circ}

iv)   10 right angles              v)  16 right angles             vi) 20 right angles

Answer:

Sum of the interior angles \displaystyle \text{No. of Sides} =   \frac{1}{2}   \Big(   \frac{\text{sum of interior angles}}{90}   +4 \Big) 
i) 540^{\circ}  \frac{1}{2} \Big( \frac{540}{90} +4 \Big)=5
ii) 1080^{\circ}  \frac{1}{2} \Big ( \frac{1080}{90} +4 \Big)=8
iii) 1980^{\circ}  \frac{1}{2} \Big ( \frac{1980}{90} +4 \Big)=13
iv) 10 right angles \frac{1}{2} \Big ( \frac{900}{90} +4 \Big)=7
v) 16 right angles \frac{1}{2} \Big ( \frac{1440}{90} +4 \Big)=10
vi) 20 right angles \frac{1}{2} \Big ( \frac{1800}{90} +4 \Big)=12

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Question 3: The sides of a hexagon are produced in order. If the measure of the exterior angles so obtained are (3x-5)^{\circ} , (8x+3)^{\circ} , (7x-2)^{\circ} , (4x+1)^{\circ} , (6x+4)^{\circ} and (2x-1)^{\circ} . Find the value of x and the measure of each exterior angle of the hexagon.

Answer:

Sum of the interior angles of hexagon = (2n-4)\times 90^{\circ} =(2\times 6-4)\times 90^{\circ}=720^{\circ}

\Rightarrow (3x-5)+(8x+3)+(7x-2)+(4x+1)+(6x+4)+ (2x-1)=360^{\circ}

\Rightarrow 30x=360^{\circ} 

\Rightarrow x=12^{\circ} 

Now substitute the value of x   in the expressions of all the sides we get:

(3x-5)^{\circ}=31^{\circ}                (8x+3)^{\circ}=99^{\circ} 

(7x-2)^{\circ}=82^{\circ}                (4x+1)^{\circ}=49^{\circ} 

(6x+4)^{\circ}= 76^{\circ}                (2x-1)^{\circ}=23^{\circ} 

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Question 4: Is it possible to have a polygon whose sum of interior angles is 840^{\circ}  

Answer:

No. Let us calculate the number of sides of this polygon.

\displaystyle \text{No. of Sides  } =   \frac{1}{2} \Big(   \frac{\text{sum of interior angles}}{90}   +4 \Big)=   \frac{1}{2}   \Big(   \frac{840^{\circ}}{90}   +4 \Big)   \text{which is not an integer.}

Hence not possible.

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Question 5: Is it possible to have a polygon, the sum of whose interior angles is 7 right angles.

Answer:

No.  Let us calculate the number of sides of this polygon.

\displaystyle \text{No. of Sides  } =   \frac{1}{2} \Big(   \frac{\text{sum of interior angles}}{90}   +4 \Big)=   \frac{1}{2}   \Big(   \frac{7 \times 90^{\circ}}{90}   +4 \Big)  = 5.5

Hence not possible.

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Question 6: Is it possible to have a polygon, the sum of whose interior angles is 14 right angles. If yes, how many sides does this polygon have?

Answer:

Yes.  Let us calculate the number of sides of this polygon.

\displaystyle \text{No. of Sides  } =   \frac{1}{2} \Big(   \frac{\text{sum of interior angles}}{90}   +4 \Big)=   \frac{1}{2}   \Big(   \frac{14 \times 90^{\circ}}{90}   +4 \Big)  = 9

Hence  possible.

The number of side =9 

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Question 7: Find the measure of each angle of a regular polygon:

i)  Pentagon       ii)  Hexagon        iii)  Heptagon       iv)  Octagon

Answer:

Polygon Sides = n Interior angles

\displaystyle =   \frac{(2n-4)\times 90^{\circ}}{n}

Exterior Angle = 180- Interior Angle
i) Pentagon 5 \displaystyle \frac{(2\times 5-4)\times 90^{\circ}}{5}  =108^{\circ} \displaystyle 180- 108=72^{\circ}
ii) Hexagon 6 \displaystyle \frac{(2\times 6-4)\times 90^{\circ}}{6}  =120^{\circ} \displaystyle 180- 120=60^{\circ}
iii) Heptagon 7 \displaystyle \frac{(2\times 7-4)\times 90^{\circ}}{7}  =  \frac{900^{\circ}}{7} 1\displaystyle 80-  \frac{900}{7}  =  \frac{360^{\circ}}{7}
iv) Octagon 8 \displaystyle \frac{(2\times 8-4)\times 90^{\circ}}{8}  =135^{\circ} \displaystyle 180- 135=65^{\circ}

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Question 8: Find the measure of each angle of a regular polygon having:

i)  9 sides     ii)  15 sides     iii)  24 sides     iv)  30 sides

Answer:

 Sides = n interior \ angles = 

\frac{(2n-4)\times 90^{\circ}}{n}

Exterior Angle = 180- Interior Angle
i) 9  \displaystyle \frac{(2\times 9-4)\times 90^{\circ}}{9}   =140^{\circ}  180- 140=40^{\circ} 
ii) 15  \displaystyle \frac{(2\times 15-4)\times 90^{\circ}}{15}   =156^{\circ} 180- 156=24^{\circ} 
iii) 24  \displaystyle \frac{(2\times 24-4)\times 90^{\circ}}{24}   = 165^{\circ} 180- 165=25^{\circ} 
iv) 30  \displaystyle \frac{(2\times 30-4)\times 90^{\circ}}{30}   =168^{\circ} 180- 168=22^{\circ} 
v) 6  \displaystyle \frac{(2\times 6-4)\times 90^{\circ}}{6}   =120^{\circ} 180- 120=60^{\circ} 
vi) 10  \displaystyle \frac{(2\times 10-4)\times 90^{\circ}}{10}   =144^{\circ} 180- 144=36^{\circ} 
vii)  20  \displaystyle \frac{(2\times 20-4)\times 90^{\circ}}{20}   =162^{\circ} 180- 162=28^{\circ} 

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Question 9: Find the number of sides of a regular polygon each of whose exterior angles are:

i)  6 sides     ii)  10 sides     iii)  15 sides     iv)  20 sides

Answer:

Exterior angle Interior angles = 180^{\circ}- Exterior Angle \displaystyle n=    \frac{360^{\circ}}{(180^{\circ}-\text{Interior Angle})} 
i) 30  180-30=150^{\circ}  \displaystyle n=   \frac{360}{180-150}     =12 
ii) 36  180-36=144^{\circ}  \displaystyle n=   \frac{360}{180-144}   =10 
iii) 40  180-40=140^{\circ}  \displaystyle n=   \frac{360}{180-140}   =9 
iv) 18  180-18=162^{\circ}  \displaystyle n=   \frac{360}{180-162}   =20 

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Question 11: Is it possible to have a regular polygon whose interior angles measure 130^{\circ}

Answer:

No.  Let us calculate the number of sides of this polygon.

\displaystyle \text{No. of Sides } n=   \frac{360^{\circ}}{(180^{\circ}-Interior Angle)} = \frac{360^{\circ}}{(180^{\circ}-130^{\circ})}   =7.2 

Hence  not possible.

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Question 12: Is it possible to have a regular polygon whose interior angles measure measures 1 \frac{3}{4}   of a right angle.

Answer:

Yes.  Let us calculate the number of sides of this polygon.

\displaystyle \text{No. of Sides } n=   \frac{360^{\circ}}{(180^{\circ}-Interior Angle)} = \frac{360^{\circ}}{(180^{\circ}-157.5^{\circ})}   =16 

Hence possible

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Question 13: Find the number of sides of a regular polygon, if its interior angle is equal to exterior angle.

Answer:

 Interior \ angles = 180^{\circ}-Exterior \ Angle

This means that each of the interior and the exterior angles  =90^{\circ}

\displaystyle \text{No. of Sides } n=   \frac{360^{\circ}}{(180^{\circ}-Interior Angle)} = \frac{360^{\circ}}{(180^{\circ}-90^{\circ})}   =4 

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Question 14: The ratio between the exterior angle and the interior angles is 2:7 . Find the number of sides of the polygon.

Answer:

 Interior \ angles = 180^{\circ}-Exterior \ Angle

Let the Exterior Angle =2x and the Interior Angle be  7x

\Rightarrow 2x+7x=180 \ or\  x=20

Therefore Interior angle =140^{\circ} 

\displaystyle \text{No. of Sides } (n)=   \frac{360^{\circ}}{180^{\circ}-140^{\circ}}   = 9 

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Question 15: The sum of all the interior angles of a regular polygon is twice the sum of exterior angles. Find the number of sides of the polygon.

Answer:

Sum of Interior Angles =(2n-4)\times 90^{\circ}

Sum of Exterior Angles =360^{\circ}

Given Sum of Interior Angles =2\times (Sum \ of \ Exterior \ Angles) 

\Rightarrow (2n-4)\times 90^{\circ} =720^{\circ} or

n=6

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Question 16: Each exterior angle of a regular polygon is (22.5)^{\circ} . Find the number of sides of the polygon.

Answer:

 Interior \ angles = 180^{\circ}-Exterior \ Angle  =180-22.5=157.5^{\circ}

\displaystyle \text{No. of Sides } n=   \frac{360^{\circ}}{(180^{\circ}-Interior Angle)} = \frac{360^{\circ}}{(180^{\circ}-157.5^{\circ})}   =16 

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Question 17: One angle of an Octagon is 100^{\circ} . And all the other seven angles are equal. What is the measure of each one of the equal angles?

Answer:

One angle given =100^{\circ}

Let each of the equal angles =x

Sum of Interior Angles =(2n-4)\times 90^{\circ} =(2\times 8-4)\times 90^{\circ}=1080^{\circ}

\Rightarrow 100+7x=1080 or x= 140^{\circ}

Each of the equal angles =140^{\circ}

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Question 18: The angles of Septagon are in the ratio of 1:2:3:4:5:6:7:8 . Find the smallest angle.

Answer:

Sum of Interior Angles =(2n-4)\times 90^{\circ}= (2\times 8-4)\times 90^{\circ}=1080^{\circ}

Let the angles be 1x, 2x, 3x, 4x, 5x, 6x, 7x, 8x

Therefore 1x+ 2x+ 3x+ 4x+ 5x+ 6x+ 7x+ 8x=1080^{\circ} or x=30

Hence the angles are 30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}, 210^{\circ} \ and \ 240^{\circ}.

The smallest angle is 30^{\circ}

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Question 19: Two angles of a polygon are right angles and each of the other angles is 120^{\circ} . Find the number of sides of the polygon.

Answer:

Let the number of sides =n

Sum of Exterior Angles =2\times 90^{\circ}+(n-2)\times 60

\Rightarrow 60+60n=360

n=5

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Question 20: Each interior angle of a regular polygon is 144 . Find the interior angle of a polygon, which has double the number of sides as the first polygon.

Answer:

Let the number of sides of the first polygon   =n

\displaystyle n=   \frac{360^{\circ}}{180^{\circ}-Interior Angle}=\frac{360^{\circ}}{180^{\circ}-144}   =10 

Therefore  the number of sides of the second  polygon   =20

\displaystyle \text{Interior angle of the second polygon }=   \frac{(2n-4)\times 90^{\circ}}{n}=  \frac{(2\times 20-4)\times 90^{\circ}}{20}   =162