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16. Understanding Shapes 
(Including Polygons) 
EXERCISE 16(A) 
Question 1. 
State which of the following are polygons : 
 
If the given figure is a polygon, name it as convex or concave. 
Solution: 
Only Fig. (ii), (iii) and (v) are polygons. 
Fig. (ii) and (iii) are concave polygons while 
Fig. (v) is convex. 
Question 2. 
Calculate the sum of angles of a polygon with : 
(i) 10 sides 
(ii) 12 sides 
(iii) 20 sides 
(iv) 25 sides 
Solution: 
(i) No. of sides n = 10 
sum of angles of polygon = (n – 2) x 180° 
= (10 – 2) x 180° = 1440° 
(ii) no. of sides n = 12 
sum of angles = (n – 2) x 180° 
= (12 – 2) x 180° = 10 x 180° = 1800° 
Page 2


16. Understanding Shapes 
(Including Polygons) 
EXERCISE 16(A) 
Question 1. 
State which of the following are polygons : 
 
If the given figure is a polygon, name it as convex or concave. 
Solution: 
Only Fig. (ii), (iii) and (v) are polygons. 
Fig. (ii) and (iii) are concave polygons while 
Fig. (v) is convex. 
Question 2. 
Calculate the sum of angles of a polygon with : 
(i) 10 sides 
(ii) 12 sides 
(iii) 20 sides 
(iv) 25 sides 
Solution: 
(i) No. of sides n = 10 
sum of angles of polygon = (n – 2) x 180° 
= (10 – 2) x 180° = 1440° 
(ii) no. of sides n = 12 
sum of angles = (n – 2) x 180° 
= (12 – 2) x 180° = 10 x 180° = 1800° 
(iii) n = 20 
Sum of angles of Polygon = (n – 2) x 180° 
= (20 – 2) x 180° = 3240° 
(iv) n = 25 
Sum of angles of polygon = (n – 2) x 180° 
= (25 – 2) x 180° = 4140° 
Question 3. 
Find the number of sides in a polygon if the sum of its interior angles is : 
(i) 900° 
(ii) 1620° 
(iii) 16 right-angles 
(iv) 32 right-angles. 
Solution: 
(i) Let no. of sides = n 
Sum of angles of polygon = 900° 
(n – 2) x 180° = 900° 
n – 2 =  
n – 2 = 5 
n = 5 + 2 
n = 7 
(ii) Let no. of sides = n 
Sum of angles of polygon = 1620° 
(n – 2) x 180° = 1620° 
n – 2 =  
n – 2 = 9 
n = 9 + 2 
n = 11 
(iii) Let no. of sides = n 
Sum of angles of polygon = 16 right angles = 16 x 90 = 1440° 
(n – 2) x 180° = 1440° 
n – 2 =  
n – 2 = 8 
n = 8 + 2 
n = 10 
(iv) Let no. of sides = n 
Sum of angles of polygon = 32 right angles = 32 x 90 = 2880° 
(n – 2) x 180° = 2880 
n – 2 =  
n – 2 = 16 
n = 16 + 2 
n = 18 
Page 3


16. Understanding Shapes 
(Including Polygons) 
EXERCISE 16(A) 
Question 1. 
State which of the following are polygons : 
 
If the given figure is a polygon, name it as convex or concave. 
Solution: 
Only Fig. (ii), (iii) and (v) are polygons. 
Fig. (ii) and (iii) are concave polygons while 
Fig. (v) is convex. 
Question 2. 
Calculate the sum of angles of a polygon with : 
(i) 10 sides 
(ii) 12 sides 
(iii) 20 sides 
(iv) 25 sides 
Solution: 
(i) No. of sides n = 10 
sum of angles of polygon = (n – 2) x 180° 
= (10 – 2) x 180° = 1440° 
(ii) no. of sides n = 12 
sum of angles = (n – 2) x 180° 
= (12 – 2) x 180° = 10 x 180° = 1800° 
(iii) n = 20 
Sum of angles of Polygon = (n – 2) x 180° 
= (20 – 2) x 180° = 3240° 
(iv) n = 25 
Sum of angles of polygon = (n – 2) x 180° 
= (25 – 2) x 180° = 4140° 
Question 3. 
Find the number of sides in a polygon if the sum of its interior angles is : 
(i) 900° 
(ii) 1620° 
(iii) 16 right-angles 
(iv) 32 right-angles. 
Solution: 
(i) Let no. of sides = n 
Sum of angles of polygon = 900° 
(n – 2) x 180° = 900° 
n – 2 =  
n – 2 = 5 
n = 5 + 2 
n = 7 
(ii) Let no. of sides = n 
Sum of angles of polygon = 1620° 
(n – 2) x 180° = 1620° 
n – 2 =  
n – 2 = 9 
n = 9 + 2 
n = 11 
(iii) Let no. of sides = n 
Sum of angles of polygon = 16 right angles = 16 x 90 = 1440° 
(n – 2) x 180° = 1440° 
n – 2 =  
n – 2 = 8 
n = 8 + 2 
n = 10 
(iv) Let no. of sides = n 
Sum of angles of polygon = 32 right angles = 32 x 90 = 2880° 
(n – 2) x 180° = 2880 
n – 2 =  
n – 2 = 16 
n = 16 + 2 
n = 18 
Question 4. 
Is it possible to have a polygon ; whose sum of interior angles is : 
(i) 870° 
(ii) 2340° 
(iii) 7 right-angles 
(iv) 4500° 
Solution: 
(i) Let no. of sides = n 
Sum of angles = 870° 
(n – 2) x 180° = 870° 
n – 2 =  
n – 2 =  
n =  + 2 
n =  
Which is not a whole number. 
Hence it is not possible to have a polygon, the sum of whose interior angles is 870° 
(ii) Let no. of sides = n 
Sum of angles = 2340° 
(n – 2) x 180° = 2340° 
n – 2 =  
n – 2 = 13 
n = 13 + 2 = 15 
Which is a whole number. 
Hence it is possible to have a polygon, the sum of whose interior angles is 2340°. 
(iii) Let no. of sides = n 
Sum of angles = 7 right angles = 7 x 90 = 630° 
(n – 2) x 180° = 630° 
n – 2 =  
n – 2 =  
n =  + 2 
n =  
Which is not a whole number. Hence it is not possible to have a polygon, the sum of 
whose interior angles is 7 right-angles. 
(iv) Let no. of sides = n 
(n – 2) x 180° = 4500° 
n – 2 =  
n – 2 = 25 
n = 25 + 2 
n = 27 
Which is a whole number. 
Hence it is possible to have a polygon, the sum of whose interior angles is 4500°. 
Page 4


16. Understanding Shapes 
(Including Polygons) 
EXERCISE 16(A) 
Question 1. 
State which of the following are polygons : 
 
If the given figure is a polygon, name it as convex or concave. 
Solution: 
Only Fig. (ii), (iii) and (v) are polygons. 
Fig. (ii) and (iii) are concave polygons while 
Fig. (v) is convex. 
Question 2. 
Calculate the sum of angles of a polygon with : 
(i) 10 sides 
(ii) 12 sides 
(iii) 20 sides 
(iv) 25 sides 
Solution: 
(i) No. of sides n = 10 
sum of angles of polygon = (n – 2) x 180° 
= (10 – 2) x 180° = 1440° 
(ii) no. of sides n = 12 
sum of angles = (n – 2) x 180° 
= (12 – 2) x 180° = 10 x 180° = 1800° 
(iii) n = 20 
Sum of angles of Polygon = (n – 2) x 180° 
= (20 – 2) x 180° = 3240° 
(iv) n = 25 
Sum of angles of polygon = (n – 2) x 180° 
= (25 – 2) x 180° = 4140° 
Question 3. 
Find the number of sides in a polygon if the sum of its interior angles is : 
(i) 900° 
(ii) 1620° 
(iii) 16 right-angles 
(iv) 32 right-angles. 
Solution: 
(i) Let no. of sides = n 
Sum of angles of polygon = 900° 
(n – 2) x 180° = 900° 
n – 2 =  
n – 2 = 5 
n = 5 + 2 
n = 7 
(ii) Let no. of sides = n 
Sum of angles of polygon = 1620° 
(n – 2) x 180° = 1620° 
n – 2 =  
n – 2 = 9 
n = 9 + 2 
n = 11 
(iii) Let no. of sides = n 
Sum of angles of polygon = 16 right angles = 16 x 90 = 1440° 
(n – 2) x 180° = 1440° 
n – 2 =  
n – 2 = 8 
n = 8 + 2 
n = 10 
(iv) Let no. of sides = n 
Sum of angles of polygon = 32 right angles = 32 x 90 = 2880° 
(n – 2) x 180° = 2880 
n – 2 =  
n – 2 = 16 
n = 16 + 2 
n = 18 
Question 4. 
Is it possible to have a polygon ; whose sum of interior angles is : 
(i) 870° 
(ii) 2340° 
(iii) 7 right-angles 
(iv) 4500° 
Solution: 
(i) Let no. of sides = n 
Sum of angles = 870° 
(n – 2) x 180° = 870° 
n – 2 =  
n – 2 =  
n =  + 2 
n =  
Which is not a whole number. 
Hence it is not possible to have a polygon, the sum of whose interior angles is 870° 
(ii) Let no. of sides = n 
Sum of angles = 2340° 
(n – 2) x 180° = 2340° 
n – 2 =  
n – 2 = 13 
n = 13 + 2 = 15 
Which is a whole number. 
Hence it is possible to have a polygon, the sum of whose interior angles is 2340°. 
(iii) Let no. of sides = n 
Sum of angles = 7 right angles = 7 x 90 = 630° 
(n – 2) x 180° = 630° 
n – 2 =  
n – 2 =  
n =  + 2 
n =  
Which is not a whole number. Hence it is not possible to have a polygon, the sum of 
whose interior angles is 7 right-angles. 
(iv) Let no. of sides = n 
(n – 2) x 180° = 4500° 
n – 2 =  
n – 2 = 25 
n = 25 + 2 
n = 27 
Which is a whole number. 
Hence it is possible to have a polygon, the sum of whose interior angles is 4500°. 
Question 5. 
(i) If all the angles of a hexagon are equal ; find the measure of each angle. 
(ii) If all the angles of a 14-sided figure are equal ; find the measure of each angle. 
Solution: 
(i) No. of sides of hexagon, n = 6 
Let each angle be = x° 
Sum of angles = 6x° 
(n – 2) x 180° = Sum of angles 
(6 – 2) x 180° = 6x° 
4 x 180 = 6x 
 
Question 6. 
Find the sum of exterior angles obtained on producing, in order, the sides of a polygon 
with : 
(i) 7 sides 
(ii) 10 sides 
(iii) 250 sides. 
Solution: 
(i) No. of sides n = 7 
Sum of interior & exterior angles at one vertex = 180° 
Page 5


16. Understanding Shapes 
(Including Polygons) 
EXERCISE 16(A) 
Question 1. 
State which of the following are polygons : 
 
If the given figure is a polygon, name it as convex or concave. 
Solution: 
Only Fig. (ii), (iii) and (v) are polygons. 
Fig. (ii) and (iii) are concave polygons while 
Fig. (v) is convex. 
Question 2. 
Calculate the sum of angles of a polygon with : 
(i) 10 sides 
(ii) 12 sides 
(iii) 20 sides 
(iv) 25 sides 
Solution: 
(i) No. of sides n = 10 
sum of angles of polygon = (n – 2) x 180° 
= (10 – 2) x 180° = 1440° 
(ii) no. of sides n = 12 
sum of angles = (n – 2) x 180° 
= (12 – 2) x 180° = 10 x 180° = 1800° 
(iii) n = 20 
Sum of angles of Polygon = (n – 2) x 180° 
= (20 – 2) x 180° = 3240° 
(iv) n = 25 
Sum of angles of polygon = (n – 2) x 180° 
= (25 – 2) x 180° = 4140° 
Question 3. 
Find the number of sides in a polygon if the sum of its interior angles is : 
(i) 900° 
(ii) 1620° 
(iii) 16 right-angles 
(iv) 32 right-angles. 
Solution: 
(i) Let no. of sides = n 
Sum of angles of polygon = 900° 
(n – 2) x 180° = 900° 
n – 2 =  
n – 2 = 5 
n = 5 + 2 
n = 7 
(ii) Let no. of sides = n 
Sum of angles of polygon = 1620° 
(n – 2) x 180° = 1620° 
n – 2 =  
n – 2 = 9 
n = 9 + 2 
n = 11 
(iii) Let no. of sides = n 
Sum of angles of polygon = 16 right angles = 16 x 90 = 1440° 
(n – 2) x 180° = 1440° 
n – 2 =  
n – 2 = 8 
n = 8 + 2 
n = 10 
(iv) Let no. of sides = n 
Sum of angles of polygon = 32 right angles = 32 x 90 = 2880° 
(n – 2) x 180° = 2880 
n – 2 =  
n – 2 = 16 
n = 16 + 2 
n = 18 
Question 4. 
Is it possible to have a polygon ; whose sum of interior angles is : 
(i) 870° 
(ii) 2340° 
(iii) 7 right-angles 
(iv) 4500° 
Solution: 
(i) Let no. of sides = n 
Sum of angles = 870° 
(n – 2) x 180° = 870° 
n – 2 =  
n – 2 =  
n =  + 2 
n =  
Which is not a whole number. 
Hence it is not possible to have a polygon, the sum of whose interior angles is 870° 
(ii) Let no. of sides = n 
Sum of angles = 2340° 
(n – 2) x 180° = 2340° 
n – 2 =  
n – 2 = 13 
n = 13 + 2 = 15 
Which is a whole number. 
Hence it is possible to have a polygon, the sum of whose interior angles is 2340°. 
(iii) Let no. of sides = n 
Sum of angles = 7 right angles = 7 x 90 = 630° 
(n – 2) x 180° = 630° 
n – 2 =  
n – 2 =  
n =  + 2 
n =  
Which is not a whole number. Hence it is not possible to have a polygon, the sum of 
whose interior angles is 7 right-angles. 
(iv) Let no. of sides = n 
(n – 2) x 180° = 4500° 
n – 2 =  
n – 2 = 25 
n = 25 + 2 
n = 27 
Which is a whole number. 
Hence it is possible to have a polygon, the sum of whose interior angles is 4500°. 
Question 5. 
(i) If all the angles of a hexagon are equal ; find the measure of each angle. 
(ii) If all the angles of a 14-sided figure are equal ; find the measure of each angle. 
Solution: 
(i) No. of sides of hexagon, n = 6 
Let each angle be = x° 
Sum of angles = 6x° 
(n – 2) x 180° = Sum of angles 
(6 – 2) x 180° = 6x° 
4 x 180 = 6x 
 
Question 6. 
Find the sum of exterior angles obtained on producing, in order, the sides of a polygon 
with : 
(i) 7 sides 
(ii) 10 sides 
(iii) 250 sides. 
Solution: 
(i) No. of sides n = 7 
Sum of interior & exterior angles at one vertex = 180° 
 
 
Question 7. 
The sides of a hexagon are produced in order. If the measures of exterior angles so 
obtained are (6x – 1)°, (10x + 2)°, (8x + 2)° (9x – 3)°, (5x + 4)° and (12x + 6)° ; find each 
exterior angle. 
Solution: 
Sum of exterior angles of hexagon formed by producing sides of order = 360°

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