Example.1) Find the magnitude of each interior angle of a regular octagon.
Ans.) The sum of the eight angle of an octagon = (2 X 8 – 4) right angles = 12 X 90⁰
Now, in a regular polygon, all the angles are equal
So, each angle of a regular octagon = (12 X 90⁰) / 8 = 135⁰ (Ans.)
Example.2) Calculate the number of sides of a regular polygon if each exterior angle is 18⁰
Ans.) The number of sides of the regular polygon
360⁰ 360⁰
= ----------------- = ----------- = 20 (Ans.)
(an exterior angle) 18⁰
Example.3) The sum of the interior angles of a polygon is 3060⁰. Then specify how many sides does it have?
Ans.) Let the number of sides of the polygon = n
Then the sum of its interior angles = (2n – 4) right angles = 3060⁰
So, (2n – 4) X 90⁰ = 3060⁰
=> 2n – 4 = 3060⁰ / 90⁰ = 34
=> 2n = 34 + 4 = 38
=> n = 38 / 2 = 19
So, n = 19 , so the polygon has 19 sides. (Ans.)
Example.4) Each interior angle of a regular polygon is 168⁰. How many sides does it have
Ans.) Each exterior angle of the regular polygon
= 180⁰ - an interior angle
= 180⁰ - 168⁰ = 12⁰
So, number of sides of the regular polygon =
360⁰ 360----------------- = ----------- = 30
An exterior angle 12
Alternatively –
If the number of sides of the regular polygon be ‘n’ then the sum of the interior angles = 168⁰ X n
= (2n – 4) X 90⁰ (by property)
=> 360⁰ = 180⁰ X n - 168⁰ X n
=> 360⁰ = 12⁰ X n
=> n = 360⁰ / 12⁰ = 30 (Ans.)
Example.5) Find the measures of each interior angle of a regular 30-gon 360⁰
Ans.) Each exterior angle of a regular 30-gon = --------- = 12⁰
30
So, each interior angle = 180⁰ - each exterior angle
= 180⁰ - 12⁰ = 168⁰ (Ans.)
Example.6) Is it possible to have a polygon in which the sum of the interior angles is 1260⁰ ?
Ans.) Let the number of sides of the polygon = n,
Given, sum of interior angles = 1260⁰
=> (2n – 4) right angle = 1260⁰
=> (2n – 4) X 90⁰ = 1260⁰
=> 2n – 4 = 1260⁰ / 90⁰ = 14
=> 2n = 14 + 4 = 18
=> n = 18 / 2 = 9
So, from the above-obtained result, we can conclude that polygon is possible. (Ans.)
Example.7) Is it possible to have a polygon in which the sum of the interior angles is 1170⁰ ?
Ans.) Let the number of sides of the polygon = n,
Given, sum of interior angles = 1170⁰
=> (2n – 4) right angle = 1170⁰
=> (2n – 4) X 90⁰ = 1170⁰
=> 2n – 4 = 1260⁰ / 90⁰ = 13
=> 2n = 13 + 4 = 17
17 1
=> n = ---------- = 8 ---------
2 2
1
A polygon cannot have 8 ----- sides.So, such a polygon cannot be possible
2
(Ans.)
Example.8) Is it possible to have a regular polygon with exterior angles of 40⁰?
360⁰
Ans.) The number of sides of the polygon = -------------------
An exterior angles
360⁰= ----------- = 9
40⁰
A polygon can have 9 sides. So, Such a regular polygon is possible. (Ans.)
Example.9) The angles of a Pentagon are in the ratio 1 : 2 : 3 : 5 : 7. find the angles
Ans.) Let the angles of the pentagon be x, 2x, 3x, 5x, and 7x
Then the sum of the angles = x + 2x + 3x + 5x + 7x = 18x
Also, the sum of the angles of the pentagon = (2n – 4) right angle
= (2 X 5 – 4) right angle
= 6 X 90⁰
As per the given condition, 18x = 6 X 90⁰
6 X 90⁰
x = ------------- = 30⁰
18
Now, x = 30⁰
2x = 2 X 30⁰ = 60⁰
3x = 3 X 30⁰ = 90⁰
5x = 5 X 30⁰ = 150⁰
7x = 7 X 30⁰ = 210⁰
Hence, the angles of the pentagon are 30⁰, 60⁰, 90⁰, 150⁰, & 210⁰ (Ans.)
Example.10) If each interior angle of a regular polygon is seventeen times an exterior angle, find the number of sides of the polygon.
Ans.) Let an exterior angles = x, then an interior angle = 17x
So, x + 17x = 180⁰
Or, 18x = 180⁰
Or, x = 10⁰
So, exterior angle 10⁰
360⁰
Number of sides of the polygon = ------------------
An exterior angle
= ------------ = 36 (Ans.)
10⁰
Example.11) Four of the angles of a heptagon are equal and the fifth is 60⁰ greater than each of the equal angles. Find the angles
Ans.) Let each of the four equal angles = x.
Then the fifth angle = x + 60⁰
Now, the sum of the angles of the heptagon = 6x + (x + 60⁰) = 7x + 60⁰
Now, as per the given condition –
7x + 60⁰ = (2n – 4) right angle
Or, 7x + 60⁰ = (2 X 7 – 4) right angle
Or, 7x + 60⁰ = (14 – 4) X 90⁰
Or, 7x + 60⁰ = 10 X 90⁰
Or, 7x + 60⁰ = 900⁰
Or, 7x = 900⁰ - 60⁰ = 840⁰
Or, x = 840⁰/7 = 120⁰
Thus, the each of the six equal angles = 120⁰ and the seventh angles
= 120⁰ + 60⁰ = 180⁰ (Ans.)
Example.12) Eight angles of a polygon are 170⁰ each. The remaining angles are 160⁰ each. Calculate the number of sides of the polygon.
Ans.) Let the number of the sides of the polygon = n
Then the sum of the angles of the polygon = 8 X 170⁰ + (n – 8) X 160⁰
and (2n – 4) right-angle = (2n – 4) X 90⁰
as per the given condition –
(2n – 4) X 90⁰ = 8 X 170⁰ + (n – 8) X 160⁰
=> 180⁰n - 360⁰ = 1360⁰ + 160⁰n - 1280⁰
=> 180⁰n - 160⁰n = 80⁰ + 360⁰
=> 20⁰n = 440⁰
=> n = 440⁰ / 20⁰ = 22
Hence the polygon has 22 sides. (Ans.)