Polygons CBSE Class 8

What is Polygon?

a plane figure with at least three straight sides and angles, and typically five or more.

Example of Polygons:

Triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon according as it contains 3,4,5,6,7,8,9,10 sides respectively.

Different types of polygons:

Concave Polygon: A polygon which has at least one angle is greater than 180° is called concave polygon.

Convex Polygon: A polygon in which all the angles are less than 180° is called convex polygon.

Regular Polygon: A polygon in which all sides and all angles are equal is called regular polygon. Example: equilateral triangle, square.

Irregular Polygon: A polygon in which all sides and all angles are not equal is called irregular polygon. Example: rectangle, rhombus.

1. Find the measure of each exterior angle of a regular  (i) triangle  (ii) 4 sides polygon  (iii) Pentagon  (iv) 10 sides polygon   (v) 15 sides polygon.

Answer:

(i) Each exterior angle of a regular triangle ( equilateral triangle) = (360/3)= 120°

(ii) Each exterior angle of a regular polygon of sides 4 = (360/4) = 90°

(iii) Each exterior angle of a regular pentagon = (360/5) = 72°

(iv) Each exterior angle of a regular polygon of sides 10 = (360/10) = 36°

(v) Each exterior angle of a regular polygon of sides 15 = (360/15) = 24°

 

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2. Is it possible to have a regular polygon each of whose exterior angles is (i) 60°  (ii) 55°?

Answer:

(i) Let the number of sides of given polygon be n

Therefore, each of its exterior angle = (360/n)°

Now, (360/n) = 60

→ n= 360/60

→ n= 6 (which is a whole number)

So, it is possible to have a regular polygon each of whose exterior angles is 60°

(ii)  Let the number of sides of given polygon be n

Therefore, each of its exterior angle = (360/n)°

Now, (360/n) = 55

→ n = (360/55)

→ n = 72/11 (which is not a whole number)

So, it is not possible to have a regular polygon each of whose exterior angles is 55°

3. Find the measure of each interior angle of a regular polygon having    (i) 10 sides  (ii) 15 sides  (iii) 8 sides.

Answer:

(i) Sum of all exterior angles = 360°

Each exterior angle = (360/10) = 36°

Therefore, each interior angle = (180-36) = 144°

(ii) Sum of all exterior angles = 360°

Each exterior angle = (360/15) = 24°

Therefore, each interior angle = (180-24) = 156°

(iii) Sum of all exterior angles = 360°

Each exterior angle = (360/8) = 45°

Therefore, each interior angle = (180-45) = 135°

(4) Is it possible to have a regular polygon each of whose interior angles is 100°?

Answer:

Each interior angle = 100°

Each exterior angle = (180-100) = 80°

Sum of all exterior angles = 360°

Therefore, Number of sides = (360/80) = (9/2)

(which is not a whole number)

It is not possible.

5. What is the sum of all interior angles of a regular  (i) pentagon  (ii) hexagon  (iii) heptagon?

Answer:

(i) Sum of all interior angles of a regular pentagon = (2  x  5  - 4) right angles

= 6 right angles

(ii) Sum of all interior angles of a regular hexagon

= (2  x  6  - 4) right angles

= 8 right angles

(iii) Sum of all interior angles of a regular heptagon

 = (2  x  7  - 4) right angles

= 10 right angles

6. What is the number of diagonals in a  (i) pentagon  (ii) hexagon  (iii) polygon of 12 sides?

Answer:

(i) Number of diagonals of a polygon of n sides = [n(n-3)/2]

Therefore, number of diagonals of a pentagon = [5(5-3)/2] = 5

(ii) number of diagonals of a pentagon = [6(6-3)/2] = 9

(iii) number of diagonals of a pentagon

 = [12(12-3)/2] = 54

7. Find the number of sides of a regular polygon whose each exterior angles measures   (i) 40°  (ii) 60°  (iii) 90°

Answer:

(i) Sum of all exterior angles of a regular polygon = 360°

Each exterior angle = 40°

Therefore, number of exterior angles = (360/40) = 9

Number of sides of that regular polygon = 9

(ii) Sum of all exterior angles of a regular polygon = 360°

Each exterior angle = 60°

Therefore, number of exterior angles = (360/60) = 6

Number of sides of that regular polygon = 6

(iii) Sum of all exterior angles of a regular polygon = 360°

Each exterior angle = 90°

Therefore, number of exterior angles = (360/90) = 4

Number of sides of that regular polygon = 4

8. In the given figure find the measure ‘x’.

Answer: Sum of all exterior angles of a polygon = 360°

Therefore, 50+115+90+x = 360

               → x = 105°

9. Find the angle measure ‘x’ in the given figure.

Answer: Sum of all interior angles of a polygon

 = (2  x  5   -  4) right angles

= 6 right angles

= 6  x  90°

Since this is a regular pentagon

Therefore, x = (6  x  90°)/ /5

               → x = 108°

10. Find the value of (x+y+z)

Answer:

Exterior angle + interior adjacent angle = 180°

X = (180 – 130) = 50°

Y = (180 – 90 ) = 90°

Z = (180 – 50) = 130°

(x+y+z) = 50+90+130 = 270°

Some Important Formulas of Polygon:

(i) Sum of all exterior angles = 360°

(ii) Each exterior angle = (360/n)° [n = number of sides]

(iii) Each interior angle = 180° - (each exterior angle)

(iv) Sum of all interior angles = (2n - 4) right angles

(v) Number of diagonals in a polygon of n sides = n(n-3)/2   

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