Polygon MCQ Quiz - Objective Question with Answer for Polygon - Download Free PDF

Given:

Angles of a pentagon are in the ratio 1 : 3 : 6 : 7 : 10.

Sum of interior angles of a pentagon = 540º.

Formula Used:

Sum of angles in a polygon = (n - 2) × 180, where n is the number of sides.

Individual angle = (Ratio of angle / Total ratio) × Sum of angles.

Calculation:

Total ratio = 1 + 3 + 6 + 7 + 10 = 27.

Smallest angle corresponds to ratio 1.

Smallest angle = (1 / 27) × 540.

⇒ Smallest angle = 540 / 27.

⇒ Smallest angle = 20º.

The smallest angle is 20º.

Concept used:

If the number of sides of a regular polygon be n, then

The interior angle = \( \frac{(n - 2) \times 180^\circ }{n}\)

Calculation:

Let the required difference be x.

No. of sides of a regular polygon = 10

According to the question

\(\Rightarrow \frac{{\left( {n - 2} \right) \times 180^\circ }}{n} - \frac{{360^\circ }}{n} = x\)

\(\Rightarrow \frac{{\left( {10 - 2} \right) \times 180^\circ }}{{10}} - \frac{{360^\circ }}{{10}} = x\)

\(\Rightarrow \frac{{8 \times 180^\circ }}{{10}} - 36^\circ = x\)

⇒ 144 - 36° = x

Given:

A regular pentagon has 5 sides, and a regular octagon has 8 sides.

The measure of the interior angle of a regular polygon is given by:

Interior angle = [(n - 2) × 180] / n, where n is the number of sides.

Formula used:

Ratio = Measure of the angle of the pentagon / Measure of the angle of the octagon

Calculations:

Step 1: Calculate the measure of the angle of the pentagon:

Angle = [(5 - 2) × 180] / 5

Angle = (3 × 180) / 5

Angle = 540 / 5

Angle = 108°

Step 2: Calculate the measure of the angle of the octagon:

Angle = [(8 - 2) × 180] / 8

Angle = (6 × 180) / 8

Angle = 1080 / 8

Angle = 135°

Step 3: Find the ratio:

Ratio = 108 / 135

Ratio = 4 / 5

The ratio of the measure of the angle of a regular pentagon to the measure of the angle of a regular octagon is 4:5.

Concept used:

1. A pentadecagon has 15 sides.

2. Number of diagonals of a polygon = n(n - 3)/2 (where n = no. of sides of the polygon)

Calculation:

Now, the number of diagonals of the pentadecagon

⇒ 15(15 - 3) ÷ 2

⇒ 90

∴ A pentadecagon has 90 diagonals.

Formula used:

Each Interior Angles = \(\rm\frac{n−2}{n}\) x 180° 

Calculation:

Let the number of sides of the polygons be 2n and n respectively.

According to the question:

18 = \(\rm\frac{2n−2}{2n}\) × 180° − \(\rm\frac{n−2}{n}\) × 180° 

18 = \(\rm\left[\frac{n−1}{n}−\frac{n−2}{n}\right]\) 180°

18 = \(\rm\frac{1}{n}\) × 180°

⇒ n = 10

∴ 2n = 20

Concept:

Octagon has eight sides.

Dodecagon has twelve sides.

Formula:

Interior angle of polygon = [(n – 2) × 180°] /n

Calculation:

Interior angle of octagon = [(8 – 2)/8] × 180° = 1080°/8 = 135°

Interior angle of dodecagon = [(12 – 2)/12] × 180° = 1800°/12 = 150°

∴ The ratio of the measures of the interior angles for octagon and dodecagon is 9 : 10

Given:

External angle = 45° 

Formula used:

External angle = (360°/n)

Number of diagonal of a n side polygon = (n² - 3n)/2

Where, n = Equal to the number of side of a polygon

Calculation:

External angle = (360°/n)

⇒ 45° = (360°/n)

⇒ n = 8 

Now, Number of diagonal of a 'n' side polygon

⇒ (n2 - 3n)/2

⇒ (64 - 24)/2

⇒ 20

∴ The number of diagonal is 20.

Each interior angle is 150

Exterior angle = 180 - 150 = 30

We know,

Exterior angle = 360°/Number of sides

⇒ Number of sides = 360°/Exterior angle = 360/30 = 12

Given:

Sum of all interior angles = 2160° 

Formula used:

Sum of interior angles of polygon = (n - 2) × 180° 

Where 'n' is the number of sides of the polygon.

Calculation:

∵ The sum of all the angles of the polygon = 2160° 

⇒ (n - 2) × 180° = 2160° 

⇒ n - 2 = 2160°/180° 

⇒ n - 2 = 12

⇒ n = 12 + 2

⇒ n = 14

Concept:

Each angle of n-sided polygon = ((n - 2) × 180)/n

Number of diagonals of n - sided polygon = n(n - 3)/2

Calculation:

Each interior angle = 150° 

150° = (n - 2) × 180)/n

⇒ 6n - 12 = 5n

n = 12 = Total side of the polygon

∴ Number of diagonals of n - sided polygon = n(n - 3)/2 = 108/2

∴ Number of diagonals of n - sided polygon = 54

Given

One of the internal angles of a regular polygon is 135°

Concept

Each interior angle of a regular polygon = [(n -2)/n] × 180° 

Number of diagonals = [n(n - 3)/2] 

Calculation

⇒ 135° = [(n -2)/n] × 180°  

⇒ (135°/180°) = [(n -2)/n]

⇒ (3/4) = [(n -2)/n]

⇒ 3n = 4n - 8

⇒ n = 8

Now, we get

⇒ Number of diagonals = 8(8 - 3)/2

⇒ Number of diagonals = 20 

∴ Number of diagonals is 20

Each interior angle of a regular polygon is 135,

⇒ Exterior angle = 180° - Interior angle = 45°

⇒ Number of sides of polygon = 360°/Exterior angle = 8

Given:

The sum of interior angles of a polygon = 1080°

Formula used:

The sum of interior angles of a polygon = (n – 2)180°

Number of diagonals = [n(n – 3)]/2

Here,

n = number of sides

Calculation:

The sum of interior angles of a polygon = 1080°

⇒ (n – 2)180° = 1080°

⇒ n – 2 = 6

⇒ n = 8

⇒ Number of diagonals = [n(n – 3)]/2

⇒ Number of diagonals = (8 × 5)/2 = 20

Given:

Regular polygon with 15 sides

Formula Used:

The sum of interior angles of polygon of n sides

= (n − 2) × 180° where n is the number of sides of polygon

Calculation:

The sum of interior angles of polygon of 15 sides

= (15 − 2) × 180° = 2340°

∴ Each interior angle of the regular polygon = 2340/15 = 156° 

Given:

The sides of the two polygons are in the ratio 4 : 5 and the ratio of their internal angles is 15 : 16

Formula used:

Internal angle of a polygon = (n - 2)/n × 180° 

Where n = number of sides of the polygon

Calculation:

Let the number of sides of the two polygons be 4x and 5x 

According to the formula,

⇒ {(4x - 2)/4x × 180°}/{(5x - 2)/5x × 180°} = 15/16

⇒ (4x - 2)/(5x - 2) × 5/4 = 15/16

⇒ (4x - 2)/(5x - 2) = 3/4

⇒ 16x - 8 = 15x - 6

⇒ x = 8 - 6 = 2

So, the number of sides of the two polygons is 8 and 10.