SSC CGL Previous Year Questions (2023 - 18): Mensuration | SSC CGL Mathematics Previous Year Paper (Topic-wise) PDF Download

 Page 1


 Pinnacle  Day: 20th - 29th  Mensuration 
 Mensuration 
 2- D:  Under 2 dimensions we will study 
 ?  Triangles 
 ?  Quadrilateral 
 ?  Polygons 
 ?  Circle 
 3 - D: Under 3 dimensions we will study 
 ?  Cube                   
 ?  Cuboids 
 ?  Box                     
 ?  Cylinder 
 ?  Prism 
 ?  Cone 
 ?  Pyramid           
 ?  Frustum 
 ?  Sphere               
 ?  Hemisphere 
 ?  Tetrahedral 
 Note:  For  2D  ?gures,  area  and  perimeter 
 are  calculated  and  for  3D  ?gures,  volume 
 and surface area are calculated. 
 TWO DIMENSIONAL FIGURES 
 TRIANGLE 
 For a triangle with height, h, and base, b; 
 Area = 
 1 
 2 
    ×     ??     ×     h 
 Perimeter = Sum of all sides 
 Special cases: 
 1)  Equilateral  Triangle  -  All  sides  are 
 equal and each angle is  .  60° 
 ?  Area  = 
 3 
 4 
 ?? 
 2 
 ?  Height  = 
 3 
 2 
 ?? 
 ?  Perimeter =  3  ?? 
 ?  Inradius(r) = 
 ?? 
 2  3 
 ?  Circumradius(R) = 
 ?? 
 3 
 If  P1,  P2  and  P3  are  perpendicular  to  a 
 side from a point inside the triangle, then 
 ?  Height of triangle = P1+P2+P3 
 2)  Scalene  Triangle  –  All  sides  will  be 
 unequal lengths. 
 Area  =  ,  ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 This  formula  is  called  Heron's 
 formula. 
 Where, Semi-perimeter, 
 =  ?? 
 ??    +    ??    +    ?? 
 2 
 ?  Perimeter  =  ?? + ?? + ?? 
 3)  Isosceles  Triangle  :-  two  sides  and 
 two  angles  are  equal.  Altitude  bisects  the 
 base. 
 ?  Area  = 
 ?? 
 4 
 4  ?? 
 2 
- ?? 
 2 
 ?  Height =  ?? 
 2 
-
 ?? 
 2 
( )
 2 
 = 
 1 
 2 
 4  ?? 
 2 
- ?? 
 2 
 ?  Perimeter =  ?? + ?? + ?? 
= 2  ?? + ?? 
 4)  Right  angled  Triangle  :-  One  of  the 
 angles is 90  .  ° 
 Here, p = perpendicular , b = base and 
 h = hypotenuse 
 ?  Area = 
 1 
 2 
    ×     ??     ×     ?? 
 ?  Perimeter  =  ?? + ?? + h 
 ?  Pythagoras Theorem: 
 h 
 2 
= ?? 
 2 
+ ?? 
 2 
 ?  Inradius(r) = 
 ?? + ?? - h 
 2 
 ?  Circumradius(R) = 
 h 
 2 
 Note:  Common Pythagoras triplets: 
 (1,  1,  )  ;  (1,  2,  )  ;  (3,  4,  5);  (5,  12,  13);  2  5 
 (7, 24, 25); (20, 21, 29), (9, 40, 41); 
 (8, 15, 17) (12, 35, 37) (60, 11, 61) 
 (65, 72 97) (96, 110, 146). 
 Try remembering them. 
 QUADRILATERAL 
 A  ?gure  enclosed  by  four  sides  is  called 
 a  quadrilateral.  A  quadrilateral  has  four 
 angles  and  sum  of  these  angles  is  equal 
 to 360  .  ° 
 Special Cases: 
 1)  Parallelogram  -  It  is  a  quadrilateral 
 with opposite sides parallel and equal. 
 ?  Area  = base x height 
 ?  Perimeter = 2 (a + b) 
 ?  +  = 2(  +  )  ?? 
 1 
 2 
 ?? 
 2 
 2 
 ?? 
 2 
 ?? 
 2 
 ?  In  a  parallelogram,  opposite 
 sides  are  equal,  opposite  angles 
 are  equal  and  diagonals  bisect 
 each other. 
 2) Rhombus  - It is a parallelogram with 
 all four sides equal. The opposite angles 
 in a rhombus are equal. 
 Here, a = side;  and  are diagonals.  ?? 
 1 
 ?? 
 2 
 ?  Area = 
 1 
 2 
   × ?? 
 1 
× ?? 
 2 
 ?  Side (a) = 
 1 
 2 
 ?? 
 1 
 2 
+ ?? 
 2 
 2 
 ?  Perimeter = 4a 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 Diagonals bisect each other 
 at right angles . 
 3)  Trapezium  –  It  is  a  quadrilateral  with 
 one pair of opposite sides parallel. 
 Here,  a  and  b  are  parallel  sides  and  h  is 
 the  height  or  perpendicular  distance 
 between a and b. 
 ?  Area = 
 1 
 2 
 ×     h??????h?? 
× ??????  ????  ????????????????  ?????????? ( )
 = 
 1 
 2 
    ×     h        ×    ( ?? + ?? )
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Page 2


 Pinnacle  Day: 20th - 29th  Mensuration 
 Mensuration 
 2- D:  Under 2 dimensions we will study 
 ?  Triangles 
 ?  Quadrilateral 
 ?  Polygons 
 ?  Circle 
 3 - D: Under 3 dimensions we will study 
 ?  Cube                   
 ?  Cuboids 
 ?  Box                     
 ?  Cylinder 
 ?  Prism 
 ?  Cone 
 ?  Pyramid           
 ?  Frustum 
 ?  Sphere               
 ?  Hemisphere 
 ?  Tetrahedral 
 Note:  For  2D  ?gures,  area  and  perimeter 
 are  calculated  and  for  3D  ?gures,  volume 
 and surface area are calculated. 
 TWO DIMENSIONAL FIGURES 
 TRIANGLE 
 For a triangle with height, h, and base, b; 
 Area = 
 1 
 2 
    ×     ??     ×     h 
 Perimeter = Sum of all sides 
 Special cases: 
 1)  Equilateral  Triangle  -  All  sides  are 
 equal and each angle is  .  60° 
 ?  Area  = 
 3 
 4 
 ?? 
 2 
 ?  Height  = 
 3 
 2 
 ?? 
 ?  Perimeter =  3  ?? 
 ?  Inradius(r) = 
 ?? 
 2  3 
 ?  Circumradius(R) = 
 ?? 
 3 
 If  P1,  P2  and  P3  are  perpendicular  to  a 
 side from a point inside the triangle, then 
 ?  Height of triangle = P1+P2+P3 
 2)  Scalene  Triangle  –  All  sides  will  be 
 unequal lengths. 
 Area  =  ,  ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 This  formula  is  called  Heron's 
 formula. 
 Where, Semi-perimeter, 
 =  ?? 
 ??    +    ??    +    ?? 
 2 
 ?  Perimeter  =  ?? + ?? + ?? 
 3)  Isosceles  Triangle  :-  two  sides  and 
 two  angles  are  equal.  Altitude  bisects  the 
 base. 
 ?  Area  = 
 ?? 
 4 
 4  ?? 
 2 
- ?? 
 2 
 ?  Height =  ?? 
 2 
-
 ?? 
 2 
( )
 2 
 = 
 1 
 2 
 4  ?? 
 2 
- ?? 
 2 
 ?  Perimeter =  ?? + ?? + ?? 
= 2  ?? + ?? 
 4)  Right  angled  Triangle  :-  One  of  the 
 angles is 90  .  ° 
 Here, p = perpendicular , b = base and 
 h = hypotenuse 
 ?  Area = 
 1 
 2 
    ×     ??     ×     ?? 
 ?  Perimeter  =  ?? + ?? + h 
 ?  Pythagoras Theorem: 
 h 
 2 
= ?? 
 2 
+ ?? 
 2 
 ?  Inradius(r) = 
 ?? + ?? - h 
 2 
 ?  Circumradius(R) = 
 h 
 2 
 Note:  Common Pythagoras triplets: 
 (1,  1,  )  ;  (1,  2,  )  ;  (3,  4,  5);  (5,  12,  13);  2  5 
 (7, 24, 25); (20, 21, 29), (9, 40, 41); 
 (8, 15, 17) (12, 35, 37) (60, 11, 61) 
 (65, 72 97) (96, 110, 146). 
 Try remembering them. 
 QUADRILATERAL 
 A  ?gure  enclosed  by  four  sides  is  called 
 a  quadrilateral.  A  quadrilateral  has  four 
 angles  and  sum  of  these  angles  is  equal 
 to 360  .  ° 
 Special Cases: 
 1)  Parallelogram  -  It  is  a  quadrilateral 
 with opposite sides parallel and equal. 
 ?  Area  = base x height 
 ?  Perimeter = 2 (a + b) 
 ?  +  = 2(  +  )  ?? 
 1 
 2 
 ?? 
 2 
 2 
 ?? 
 2 
 ?? 
 2 
 ?  In  a  parallelogram,  opposite 
 sides  are  equal,  opposite  angles 
 are  equal  and  diagonals  bisect 
 each other. 
 2) Rhombus  - It is a parallelogram with 
 all four sides equal. The opposite angles 
 in a rhombus are equal. 
 Here, a = side;  and  are diagonals.  ?? 
 1 
 ?? 
 2 
 ?  Area = 
 1 
 2 
   × ?? 
 1 
× ?? 
 2 
 ?  Side (a) = 
 1 
 2 
 ?? 
 1 
 2 
+ ?? 
 2 
 2 
 ?  Perimeter = 4a 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 Diagonals bisect each other 
 at right angles . 
 3)  Trapezium  –  It  is  a  quadrilateral  with 
 one pair of opposite sides parallel. 
 Here,  a  and  b  are  parallel  sides  and  h  is 
 the  height  or  perpendicular  distance 
 between a and b. 
 ?  Area = 
 1 
 2 
 ×     h??????h?? 
× ??????  ????  ????????????????  ?????????? ( )
 = 
 1 
 2 
    ×     h        ×    ( ?? + ?? )
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 Pinnacle  Day: 20th - 29th  Mensuration 
 4)  Rectangle  – It is a parallelogram with 
 equal opposite sides and each angle is 90  ° 
 ?  Area = Length  Breadth = L  B × ×
 ?  Perimeter = 2 (L + B) 
 ?  Diagonal (d) =  ?? 
 2 
+ ?? 
 2 
 ?  Area  of  the  walls  of  a 
 rectangular room 
 =  2     ×    ( ?? + ?? )    ×     ?? 
 5)  Square  –  It  is  a  parallelogram  with  all 
 four  sides  equal  and  each  angle  is  equal 
 to 90  .  ° 
 ?  Area =  = ( ???????? )
 2 
= ?? 
 2 
   
 1 
 2 
 ?? 
 2 
 ?  Perimeter = 4  side = 4a ×
 ?  Diagonal (d) = a  2 
 NOTE:  Important  points  about 
 Quadrilaterals 
 1. The diagonals of a parallelogram 
 bisect each other. 
 2. Diagonal of a parallelogram divides it 
 into two triangles of equal area. 
 3. The diagonals of a rectangle are of 
 equal lengths and bisect each other. 
 4. The diagonals of a square are equal 
 and bisect each other at right angles. 
 5. A rhombus has unequal diagonals and 
 they bisect each other at right angles. 
 6. A parallelogram and a rectangle have 
 equal areas if they are on the same 
 base and between the same parallel 
 Lines. 
 REGULAR POLYGON 
 In  a  regular  polygon  all  sides  and  all 
 interior  angles  are  equal.  A  polygon  is 
 called  a  pentagon,  hexagon  ,  heptagon  , 
 octagon,  nonagon  and  a  decagon  as  they 
 have 5, 6, 7 , 8, 9, 10 sides, respectively. 
 If  each  side  of  a  regular  polygon  of  ‘n’ 
 sides is equal to ‘a’ then: 
 ?  Area of regular pentagon 
 = 
 6 . 9 
 4 
 ×  ?? 
 2 
 ?  Area of regular hexagon 
 =  6  × 
 3 
 4 
 ?? 
 2 
 ?  Each exterior angle  = 
 360° 
 ?? 
 ?  Each interior angle 
 =  180  Exterior angle  ° -   
 ?  Number of diagonals 
 = 
 ??  ??    -    3 ( )
 2 
 ?  Sum of all interior angles 
 = ( ?? - 2 ) ×180 
 ?  Sum of all exterior angle = 360° 
 CIRCLE 
 It  is  a  plane  ?gure  enclosed  by  a  line  on 
 which  every  point  is  equally  distant  from 
 a ?xed point (centre) inside the circle. 
 ?  Area = p  ?? 
 2 
 ?  Circumference  (perimeter ) 
 =  2p  ?? 
 ?  Diameter  = 2r 
 ?  Length of Arc  (AB) 
 =  2p  ??     ×    
?
 360° 
 ?  Area of sector AOB 
 = p ?? 
 2 
×
?
 360° 
 ?  Length  of  Arc  (AB)  =  ×  ?????????? 
 , (where angle in radian)  ???????????? 
 CIRCULAR RING 
 Here, R = radius of bigger ring , 
 r = radius of smaller ring 
 ?  Area =  p ( ?? 
 2 
- ?? 
 2 
)
 ?  Difference  in  circumference  of 
 both the rings =  2p  ?? - 2p  ?? 
 Short Cut methods/Tricks 
 1. If the length and breadth of a rectangle 
 are increased by a% and b%, the area 
 of the rectangle will be increased by 
 (  +  )%  ?? + ?? 
 ???? 
 100 
 2. If any of the two sides of the 
 rectangle is decreased then use ‘–ve’ 
 values  for that side. 
 3. All the sides of any two dimensional 
 ?gure changed by a%,  then its area will 
 change by (2  a  +  )%    
 ?? 
 2 
 100 
 Whenever there is a decrease, use 
 negative value for ‘a’ 
 4. If all the sides of any two dimensional 
 ?gure has changed (increased or 
 decreased) by a% then its perimeter 
 also changes by a%. In the case of a 
 circle such changes take place 
 because of the change in radius (or 
 diameter). 
 5. If the area of a square is ‘a’ square 
 unit. Then the area of the circle formed 
 with the same perimeter is given by 
 square units. 
 6. Area of a square inscribed in a 
 circle of radius ‘r’ is equal to  .  2  ?? 
 2 
 7. The area of the largest triangle 
 inscribed in a semicircle of radius r is 
 equal to  .  ?? 
 2 
 8. If a pathway of width x is made inside 
 or outside a rectangular plot of length 
 L and breath B, then area of the 
 pathway is 
 (i)  , if path is  2  ?? ( ?? + ?? + 2  ?? )
 made outside the plot 
 (ii)  , if a path is  2  ?? ( ?? + ?? - 2  ?? )
 made inside the plot. 
 9.  If  two  paths,  each  of  width  x  are  made 
 parallel  to  length  (L)  and  breadth  (B)  of 
 the  rectangular  plot  in  the  middle  of  the 
 plot crossing each other, then, 
 Area of the path  =  ?? ( ?? + ?? - ?? )
 THREE DIMENSIONAL FIGURES 
 CUBE 
 All  sides  are  equal.  It  has  six  faces  and 
 12 edges. 
 ?  Volume =  ?? 
 3 
 ?  Total surface area  = 6  ?? 
 2 
 ?  Diagonal =  ??  3 
 ?  Sum of all edges = 12a 
 Here, a = length of the side 
 CUBOID 
 A  rectangular  body  having  3D  rectangular 
 shape, is called a cuboid. 
 ?  Volume  =  ??     ×     ??     ×     h 
 ?  Total surface area  = 
 2 ( ???? + ??h + ??h )
 ?  Diagonal =  ?? 
 2 
+ ?? 
 2 
+ h 
 2 
 BOX 
 A box has its shape like a cube or cuboid. 
 The amount that a box can hold or 
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Page 3


 Pinnacle  Day: 20th - 29th  Mensuration 
 Mensuration 
 2- D:  Under 2 dimensions we will study 
 ?  Triangles 
 ?  Quadrilateral 
 ?  Polygons 
 ?  Circle 
 3 - D: Under 3 dimensions we will study 
 ?  Cube                   
 ?  Cuboids 
 ?  Box                     
 ?  Cylinder 
 ?  Prism 
 ?  Cone 
 ?  Pyramid           
 ?  Frustum 
 ?  Sphere               
 ?  Hemisphere 
 ?  Tetrahedral 
 Note:  For  2D  ?gures,  area  and  perimeter 
 are  calculated  and  for  3D  ?gures,  volume 
 and surface area are calculated. 
 TWO DIMENSIONAL FIGURES 
 TRIANGLE 
 For a triangle with height, h, and base, b; 
 Area = 
 1 
 2 
    ×     ??     ×     h 
 Perimeter = Sum of all sides 
 Special cases: 
 1)  Equilateral  Triangle  -  All  sides  are 
 equal and each angle is  .  60° 
 ?  Area  = 
 3 
 4 
 ?? 
 2 
 ?  Height  = 
 3 
 2 
 ?? 
 ?  Perimeter =  3  ?? 
 ?  Inradius(r) = 
 ?? 
 2  3 
 ?  Circumradius(R) = 
 ?? 
 3 
 If  P1,  P2  and  P3  are  perpendicular  to  a 
 side from a point inside the triangle, then 
 ?  Height of triangle = P1+P2+P3 
 2)  Scalene  Triangle  –  All  sides  will  be 
 unequal lengths. 
 Area  =  ,  ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 This  formula  is  called  Heron's 
 formula. 
 Where, Semi-perimeter, 
 =  ?? 
 ??    +    ??    +    ?? 
 2 
 ?  Perimeter  =  ?? + ?? + ?? 
 3)  Isosceles  Triangle  :-  two  sides  and 
 two  angles  are  equal.  Altitude  bisects  the 
 base. 
 ?  Area  = 
 ?? 
 4 
 4  ?? 
 2 
- ?? 
 2 
 ?  Height =  ?? 
 2 
-
 ?? 
 2 
( )
 2 
 = 
 1 
 2 
 4  ?? 
 2 
- ?? 
 2 
 ?  Perimeter =  ?? + ?? + ?? 
= 2  ?? + ?? 
 4)  Right  angled  Triangle  :-  One  of  the 
 angles is 90  .  ° 
 Here, p = perpendicular , b = base and 
 h = hypotenuse 
 ?  Area = 
 1 
 2 
    ×     ??     ×     ?? 
 ?  Perimeter  =  ?? + ?? + h 
 ?  Pythagoras Theorem: 
 h 
 2 
= ?? 
 2 
+ ?? 
 2 
 ?  Inradius(r) = 
 ?? + ?? - h 
 2 
 ?  Circumradius(R) = 
 h 
 2 
 Note:  Common Pythagoras triplets: 
 (1,  1,  )  ;  (1,  2,  )  ;  (3,  4,  5);  (5,  12,  13);  2  5 
 (7, 24, 25); (20, 21, 29), (9, 40, 41); 
 (8, 15, 17) (12, 35, 37) (60, 11, 61) 
 (65, 72 97) (96, 110, 146). 
 Try remembering them. 
 QUADRILATERAL 
 A  ?gure  enclosed  by  four  sides  is  called 
 a  quadrilateral.  A  quadrilateral  has  four 
 angles  and  sum  of  these  angles  is  equal 
 to 360  .  ° 
 Special Cases: 
 1)  Parallelogram  -  It  is  a  quadrilateral 
 with opposite sides parallel and equal. 
 ?  Area  = base x height 
 ?  Perimeter = 2 (a + b) 
 ?  +  = 2(  +  )  ?? 
 1 
 2 
 ?? 
 2 
 2 
 ?? 
 2 
 ?? 
 2 
 ?  In  a  parallelogram,  opposite 
 sides  are  equal,  opposite  angles 
 are  equal  and  diagonals  bisect 
 each other. 
 2) Rhombus  - It is a parallelogram with 
 all four sides equal. The opposite angles 
 in a rhombus are equal. 
 Here, a = side;  and  are diagonals.  ?? 
 1 
 ?? 
 2 
 ?  Area = 
 1 
 2 
   × ?? 
 1 
× ?? 
 2 
 ?  Side (a) = 
 1 
 2 
 ?? 
 1 
 2 
+ ?? 
 2 
 2 
 ?  Perimeter = 4a 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 Diagonals bisect each other 
 at right angles . 
 3)  Trapezium  –  It  is  a  quadrilateral  with 
 one pair of opposite sides parallel. 
 Here,  a  and  b  are  parallel  sides  and  h  is 
 the  height  or  perpendicular  distance 
 between a and b. 
 ?  Area = 
 1 
 2 
 ×     h??????h?? 
× ??????  ????  ????????????????  ?????????? ( )
 = 
 1 
 2 
    ×     h        ×    ( ?? + ?? )
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 Pinnacle  Day: 20th - 29th  Mensuration 
 4)  Rectangle  – It is a parallelogram with 
 equal opposite sides and each angle is 90  ° 
 ?  Area = Length  Breadth = L  B × ×
 ?  Perimeter = 2 (L + B) 
 ?  Diagonal (d) =  ?? 
 2 
+ ?? 
 2 
 ?  Area  of  the  walls  of  a 
 rectangular room 
 =  2     ×    ( ?? + ?? )    ×     ?? 
 5)  Square  –  It  is  a  parallelogram  with  all 
 four  sides  equal  and  each  angle  is  equal 
 to 90  .  ° 
 ?  Area =  = ( ???????? )
 2 
= ?? 
 2 
   
 1 
 2 
 ?? 
 2 
 ?  Perimeter = 4  side = 4a ×
 ?  Diagonal (d) = a  2 
 NOTE:  Important  points  about 
 Quadrilaterals 
 1. The diagonals of a parallelogram 
 bisect each other. 
 2. Diagonal of a parallelogram divides it 
 into two triangles of equal area. 
 3. The diagonals of a rectangle are of 
 equal lengths and bisect each other. 
 4. The diagonals of a square are equal 
 and bisect each other at right angles. 
 5. A rhombus has unequal diagonals and 
 they bisect each other at right angles. 
 6. A parallelogram and a rectangle have 
 equal areas if they are on the same 
 base and between the same parallel 
 Lines. 
 REGULAR POLYGON 
 In  a  regular  polygon  all  sides  and  all 
 interior  angles  are  equal.  A  polygon  is 
 called  a  pentagon,  hexagon  ,  heptagon  , 
 octagon,  nonagon  and  a  decagon  as  they 
 have 5, 6, 7 , 8, 9, 10 sides, respectively. 
 If  each  side  of  a  regular  polygon  of  ‘n’ 
 sides is equal to ‘a’ then: 
 ?  Area of regular pentagon 
 = 
 6 . 9 
 4 
 ×  ?? 
 2 
 ?  Area of regular hexagon 
 =  6  × 
 3 
 4 
 ?? 
 2 
 ?  Each exterior angle  = 
 360° 
 ?? 
 ?  Each interior angle 
 =  180  Exterior angle  ° -   
 ?  Number of diagonals 
 = 
 ??  ??    -    3 ( )
 2 
 ?  Sum of all interior angles 
 = ( ?? - 2 ) ×180 
 ?  Sum of all exterior angle = 360° 
 CIRCLE 
 It  is  a  plane  ?gure  enclosed  by  a  line  on 
 which  every  point  is  equally  distant  from 
 a ?xed point (centre) inside the circle. 
 ?  Area = p  ?? 
 2 
 ?  Circumference  (perimeter ) 
 =  2p  ?? 
 ?  Diameter  = 2r 
 ?  Length of Arc  (AB) 
 =  2p  ??     ×    
?
 360° 
 ?  Area of sector AOB 
 = p ?? 
 2 
×
?
 360° 
 ?  Length  of  Arc  (AB)  =  ×  ?????????? 
 , (where angle in radian)  ???????????? 
 CIRCULAR RING 
 Here, R = radius of bigger ring , 
 r = radius of smaller ring 
 ?  Area =  p ( ?? 
 2 
- ?? 
 2 
)
 ?  Difference  in  circumference  of 
 both the rings =  2p  ?? - 2p  ?? 
 Short Cut methods/Tricks 
 1. If the length and breadth of a rectangle 
 are increased by a% and b%, the area 
 of the rectangle will be increased by 
 (  +  )%  ?? + ?? 
 ???? 
 100 
 2. If any of the two sides of the 
 rectangle is decreased then use ‘–ve’ 
 values  for that side. 
 3. All the sides of any two dimensional 
 ?gure changed by a%,  then its area will 
 change by (2  a  +  )%    
 ?? 
 2 
 100 
 Whenever there is a decrease, use 
 negative value for ‘a’ 
 4. If all the sides of any two dimensional 
 ?gure has changed (increased or 
 decreased) by a% then its perimeter 
 also changes by a%. In the case of a 
 circle such changes take place 
 because of the change in radius (or 
 diameter). 
 5. If the area of a square is ‘a’ square 
 unit. Then the area of the circle formed 
 with the same perimeter is given by 
 square units. 
 6. Area of a square inscribed in a 
 circle of radius ‘r’ is equal to  .  2  ?? 
 2 
 7. The area of the largest triangle 
 inscribed in a semicircle of radius r is 
 equal to  .  ?? 
 2 
 8. If a pathway of width x is made inside 
 or outside a rectangular plot of length 
 L and breath B, then area of the 
 pathway is 
 (i)  , if path is  2  ?? ( ?? + ?? + 2  ?? )
 made outside the plot 
 (ii)  , if a path is  2  ?? ( ?? + ?? - 2  ?? )
 made inside the plot. 
 9.  If  two  paths,  each  of  width  x  are  made 
 parallel  to  length  (L)  and  breadth  (B)  of 
 the  rectangular  plot  in  the  middle  of  the 
 plot crossing each other, then, 
 Area of the path  =  ?? ( ?? + ?? - ?? )
 THREE DIMENSIONAL FIGURES 
 CUBE 
 All  sides  are  equal.  It  has  six  faces  and 
 12 edges. 
 ?  Volume =  ?? 
 3 
 ?  Total surface area  = 6  ?? 
 2 
 ?  Diagonal =  ??  3 
 ?  Sum of all edges = 12a 
 Here, a = length of the side 
 CUBOID 
 A  rectangular  body  having  3D  rectangular 
 shape, is called a cuboid. 
 ?  Volume  =  ??     ×     ??     ×     h 
 ?  Total surface area  = 
 2 ( ???? + ??h + ??h )
 ?  Diagonal =  ?? 
 2 
+ ?? 
 2 
+ h 
 2 
 BOX 
 A box has its shape like a cube or cuboid. 
 The amount that a box can hold or 
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 Pinnacle  Day: 20th - 29th  Mensuration 
 contain is called the capacity of the box. 
 Capacity means internal volume. 
 ?  Surface area of an open box 
 =  2  (  length  +  breadth  )  height ×
 + length  breadth    ×   
 =  2  ?? + ?? ( )    ×     h + ??  ×  ?? 
 ?  Capacity of box 
 =  ; ( ?? - 2  ?? )   ( ?? - 2  ?? )( h - 2  ?? )
 where, t = thickness of box 
 ?  Volume  of  the  material  of  the 
 box  =  External  volume  –  Internal 
 volume(  or  capacity)  = 
 ????h - ( ?? - 2  ?? )( ?? - 2  ?? )
( h - 2  ?? )
 ROOM 
 A  rectangular  room  has  four  walls 
 (surfaces)  and  opposite  walls  have  equal 
 area. 
 ?  Total Area of walls 
 =  2 ( ?? + ?? )    ×     h 
 ?  Total volume of the room 
 =  ??     ×     ??     ×     h 
 ?  Area of ?oor or roof =  ??     ×     ?? 
 CYLINDER 
 ?  Volume  of  cylinder  =  area  of 
 base  height = × p ?? 
 2 
 h 
 ?  Curved  surface  area  =  Perimeter 
 of base  height  = × 2p  ??h 
 ?  Total  surface  Area  =  curved 
 surface  area  +  area  of  both  the 
 circles 
 =  2p  ??h + 2p  ?? 
 2 
= 2p  ?? + h ( ) ?? 
 ?  When  the  rectangular  sheet  is 
 folded  along  its  length,  then  the 
 length  becomes  the 
 circumference  of  the  base  of 
 the  cylinder  and  breadth 
 becomes  the  height  of  the 
 cylinder. 
 HOLLOW CYLINDER 
 ?  Volume of hollow cylinder 
 = p( ?? 
 2 
- ?? 
 2 
) h 
 ?  Curved surface area 
 =  2p ( ?? + ?? ) h 
 ?  Total surface area 
 =  2p  ?? + ?? ( ) h + 2p ( ?? 
 2 
- ?? 
 2 
)
 =  2 p( ?? + ?? ) h + ?? - ?? { }
 Where,  R  =  External  radius  of 
 cylinder  ,  r  =  internal  radius  of 
 cylinder , h = height 
 PRISM 
 ?  Volume of prism 
 = area of base  height ×
 ?  Lateral  surface  area  =  Perimeter 
 of base  height ×
 ?  Total  surface  Area  =  Lateral 
 surface  area  +  area  of  base  and 
 top surface 
 CONE 
 A  solid  and  round  body  with  a  round  base 
 and pointed peak. 
 ?  Volume  =  × 
 1 
 3 
 ????????     ????????  × 
 height  = 
 1 
 3 
p ?? 
 2 
 h 
 ?  Slant height  (  ) =  ??  ?? 
 2 
+ h 
 2 
 ?  Curved surface area =  p  ???? 
= p  ??  ?? 
 2 
+ h 
 2 
 ?  Total  surface  area  =  p  ???? + p  ?? 
 2 
= p  ?? ( ?? + ?? )
 ?  Cone  formed  by  rotating  right 
 angled triangle about its height: 
 ?  Volume of cone so formed 
 = 
 1 
 3 
p ?? 
 2 
 ?? 
 ?  Similarly,  Cone  formed  by 
 rotating  right  angled  triangle 
 about its base: 
 Volume of cone so formed 
 = 
 1 
 3 
p ?? 
 2 
 ?? 
 ?  Similarly,  Cone  formed  by 
 rotating  right  angled  triangle 
 about its hypotenuse : 
 Volume of cone so formed 
 =  ,  (where  r  is  the  altitude 
 1 
 3 
p ?? 
 2 
 ?? 
 on hypotenuse and r =  ) 
 ??    ×    ?? 
 ?? 
 Note:  If  the  base  is  not  round,  it  will  be 
 called  a  pyramid.  A  pyramid  can  have 
 various  shapes  of  the  base  example: 
 square, rectangular, triangular etc. 
 PYRAMID 
 ?  Pyramid  means  a  structure  with 
 regular  polygon  as  its  base  and 
 sloping  sides  that  meet  in  a 
 point at the top. 
 ?  In  Pyramid,  with  n  sided  regular 
 polygon  at  its  base,  total 
 number of vertices = n + 1 
 ?  Volume =  ×base area × height 
 1 
 3 
 ?  Slant height  (l) =  ?? 
 2 
+ h 
 2 
 ?  Lateral surface area 
 = 
 ??????????????????     ×     ??????????     h??????h?? 
 2 
 ?  Total  Surface  area  =  Lateral 
 surface area + Area of base 
 TETRAHEDRON 
 It  is  a  3D  ?gure  made  by  joining  four 
 equilateral triangles. 
 ?  Volume  (V) 
 = 
 1 
 3 
    ×     ????????     ????????     ×     h??????h?? 
 V = ?   
 1 
 3 
×
 3 
 4 
 ?? 
 2 
×
 6 
 12 
 ?? +
 6 
 4 
 ?? 
( )
 V =  (Remember  this formula) ?
 2 
 12 
 ?? 
 3 
 ?  Lateral surface area =  a 
 2 
 3     × 
 3 
 4 
 ?  Total surface area 
 =  4 ×  a 
 2 
   
 3 
 4 
 ?  Height of Tetrahedron = 
 6 
 3 
 ?? 
 FRUSTUM OF CONE 
 If  a  cone  is  cut  by  a  plane  parallel  to  its 
 base,  so  as  to  divide  the  cone  into  two 
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Page 4


 Pinnacle  Day: 20th - 29th  Mensuration 
 Mensuration 
 2- D:  Under 2 dimensions we will study 
 ?  Triangles 
 ?  Quadrilateral 
 ?  Polygons 
 ?  Circle 
 3 - D: Under 3 dimensions we will study 
 ?  Cube                   
 ?  Cuboids 
 ?  Box                     
 ?  Cylinder 
 ?  Prism 
 ?  Cone 
 ?  Pyramid           
 ?  Frustum 
 ?  Sphere               
 ?  Hemisphere 
 ?  Tetrahedral 
 Note:  For  2D  ?gures,  area  and  perimeter 
 are  calculated  and  for  3D  ?gures,  volume 
 and surface area are calculated. 
 TWO DIMENSIONAL FIGURES 
 TRIANGLE 
 For a triangle with height, h, and base, b; 
 Area = 
 1 
 2 
    ×     ??     ×     h 
 Perimeter = Sum of all sides 
 Special cases: 
 1)  Equilateral  Triangle  -  All  sides  are 
 equal and each angle is  .  60° 
 ?  Area  = 
 3 
 4 
 ?? 
 2 
 ?  Height  = 
 3 
 2 
 ?? 
 ?  Perimeter =  3  ?? 
 ?  Inradius(r) = 
 ?? 
 2  3 
 ?  Circumradius(R) = 
 ?? 
 3 
 If  P1,  P2  and  P3  are  perpendicular  to  a 
 side from a point inside the triangle, then 
 ?  Height of triangle = P1+P2+P3 
 2)  Scalene  Triangle  –  All  sides  will  be 
 unequal lengths. 
 Area  =  ,  ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 This  formula  is  called  Heron's 
 formula. 
 Where, Semi-perimeter, 
 =  ?? 
 ??    +    ??    +    ?? 
 2 
 ?  Perimeter  =  ?? + ?? + ?? 
 3)  Isosceles  Triangle  :-  two  sides  and 
 two  angles  are  equal.  Altitude  bisects  the 
 base. 
 ?  Area  = 
 ?? 
 4 
 4  ?? 
 2 
- ?? 
 2 
 ?  Height =  ?? 
 2 
-
 ?? 
 2 
( )
 2 
 = 
 1 
 2 
 4  ?? 
 2 
- ?? 
 2 
 ?  Perimeter =  ?? + ?? + ?? 
= 2  ?? + ?? 
 4)  Right  angled  Triangle  :-  One  of  the 
 angles is 90  .  ° 
 Here, p = perpendicular , b = base and 
 h = hypotenuse 
 ?  Area = 
 1 
 2 
    ×     ??     ×     ?? 
 ?  Perimeter  =  ?? + ?? + h 
 ?  Pythagoras Theorem: 
 h 
 2 
= ?? 
 2 
+ ?? 
 2 
 ?  Inradius(r) = 
 ?? + ?? - h 
 2 
 ?  Circumradius(R) = 
 h 
 2 
 Note:  Common Pythagoras triplets: 
 (1,  1,  )  ;  (1,  2,  )  ;  (3,  4,  5);  (5,  12,  13);  2  5 
 (7, 24, 25); (20, 21, 29), (9, 40, 41); 
 (8, 15, 17) (12, 35, 37) (60, 11, 61) 
 (65, 72 97) (96, 110, 146). 
 Try remembering them. 
 QUADRILATERAL 
 A  ?gure  enclosed  by  four  sides  is  called 
 a  quadrilateral.  A  quadrilateral  has  four 
 angles  and  sum  of  these  angles  is  equal 
 to 360  .  ° 
 Special Cases: 
 1)  Parallelogram  -  It  is  a  quadrilateral 
 with opposite sides parallel and equal. 
 ?  Area  = base x height 
 ?  Perimeter = 2 (a + b) 
 ?  +  = 2(  +  )  ?? 
 1 
 2 
 ?? 
 2 
 2 
 ?? 
 2 
 ?? 
 2 
 ?  In  a  parallelogram,  opposite 
 sides  are  equal,  opposite  angles 
 are  equal  and  diagonals  bisect 
 each other. 
 2) Rhombus  - It is a parallelogram with 
 all four sides equal. The opposite angles 
 in a rhombus are equal. 
 Here, a = side;  and  are diagonals.  ?? 
 1 
 ?? 
 2 
 ?  Area = 
 1 
 2 
   × ?? 
 1 
× ?? 
 2 
 ?  Side (a) = 
 1 
 2 
 ?? 
 1 
 2 
+ ?? 
 2 
 2 
 ?  Perimeter = 4a 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 Diagonals bisect each other 
 at right angles . 
 3)  Trapezium  –  It  is  a  quadrilateral  with 
 one pair of opposite sides parallel. 
 Here,  a  and  b  are  parallel  sides  and  h  is 
 the  height  or  perpendicular  distance 
 between a and b. 
 ?  Area = 
 1 
 2 
 ×     h??????h?? 
× ??????  ????  ????????????????  ?????????? ( )
 = 
 1 
 2 
    ×     h        ×    ( ?? + ?? )
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 4)  Rectangle  – It is a parallelogram with 
 equal opposite sides and each angle is 90  ° 
 ?  Area = Length  Breadth = L  B × ×
 ?  Perimeter = 2 (L + B) 
 ?  Diagonal (d) =  ?? 
 2 
+ ?? 
 2 
 ?  Area  of  the  walls  of  a 
 rectangular room 
 =  2     ×    ( ?? + ?? )    ×     ?? 
 5)  Square  –  It  is  a  parallelogram  with  all 
 four  sides  equal  and  each  angle  is  equal 
 to 90  .  ° 
 ?  Area =  = ( ???????? )
 2 
= ?? 
 2 
   
 1 
 2 
 ?? 
 2 
 ?  Perimeter = 4  side = 4a ×
 ?  Diagonal (d) = a  2 
 NOTE:  Important  points  about 
 Quadrilaterals 
 1. The diagonals of a parallelogram 
 bisect each other. 
 2. Diagonal of a parallelogram divides it 
 into two triangles of equal area. 
 3. The diagonals of a rectangle are of 
 equal lengths and bisect each other. 
 4. The diagonals of a square are equal 
 and bisect each other at right angles. 
 5. A rhombus has unequal diagonals and 
 they bisect each other at right angles. 
 6. A parallelogram and a rectangle have 
 equal areas if they are on the same 
 base and between the same parallel 
 Lines. 
 REGULAR POLYGON 
 In  a  regular  polygon  all  sides  and  all 
 interior  angles  are  equal.  A  polygon  is 
 called  a  pentagon,  hexagon  ,  heptagon  , 
 octagon,  nonagon  and  a  decagon  as  they 
 have 5, 6, 7 , 8, 9, 10 sides, respectively. 
 If  each  side  of  a  regular  polygon  of  ‘n’ 
 sides is equal to ‘a’ then: 
 ?  Area of regular pentagon 
 = 
 6 . 9 
 4 
 ×  ?? 
 2 
 ?  Area of regular hexagon 
 =  6  × 
 3 
 4 
 ?? 
 2 
 ?  Each exterior angle  = 
 360° 
 ?? 
 ?  Each interior angle 
 =  180  Exterior angle  ° -   
 ?  Number of diagonals 
 = 
 ??  ??    -    3 ( )
 2 
 ?  Sum of all interior angles 
 = ( ?? - 2 ) ×180 
 ?  Sum of all exterior angle = 360° 
 CIRCLE 
 It  is  a  plane  ?gure  enclosed  by  a  line  on 
 which  every  point  is  equally  distant  from 
 a ?xed point (centre) inside the circle. 
 ?  Area = p  ?? 
 2 
 ?  Circumference  (perimeter ) 
 =  2p  ?? 
 ?  Diameter  = 2r 
 ?  Length of Arc  (AB) 
 =  2p  ??     ×    
?
 360° 
 ?  Area of sector AOB 
 = p ?? 
 2 
×
?
 360° 
 ?  Length  of  Arc  (AB)  =  ×  ?????????? 
 , (where angle in radian)  ???????????? 
 CIRCULAR RING 
 Here, R = radius of bigger ring , 
 r = radius of smaller ring 
 ?  Area =  p ( ?? 
 2 
- ?? 
 2 
)
 ?  Difference  in  circumference  of 
 both the rings =  2p  ?? - 2p  ?? 
 Short Cut methods/Tricks 
 1. If the length and breadth of a rectangle 
 are increased by a% and b%, the area 
 of the rectangle will be increased by 
 (  +  )%  ?? + ?? 
 ???? 
 100 
 2. If any of the two sides of the 
 rectangle is decreased then use ‘–ve’ 
 values  for that side. 
 3. All the sides of any two dimensional 
 ?gure changed by a%,  then its area will 
 change by (2  a  +  )%    
 ?? 
 2 
 100 
 Whenever there is a decrease, use 
 negative value for ‘a’ 
 4. If all the sides of any two dimensional 
 ?gure has changed (increased or 
 decreased) by a% then its perimeter 
 also changes by a%. In the case of a 
 circle such changes take place 
 because of the change in radius (or 
 diameter). 
 5. If the area of a square is ‘a’ square 
 unit. Then the area of the circle formed 
 with the same perimeter is given by 
 square units. 
 6. Area of a square inscribed in a 
 circle of radius ‘r’ is equal to  .  2  ?? 
 2 
 7. The area of the largest triangle 
 inscribed in a semicircle of radius r is 
 equal to  .  ?? 
 2 
 8. If a pathway of width x is made inside 
 or outside a rectangular plot of length 
 L and breath B, then area of the 
 pathway is 
 (i)  , if path is  2  ?? ( ?? + ?? + 2  ?? )
 made outside the plot 
 (ii)  , if a path is  2  ?? ( ?? + ?? - 2  ?? )
 made inside the plot. 
 9.  If  two  paths,  each  of  width  x  are  made 
 parallel  to  length  (L)  and  breadth  (B)  of 
 the  rectangular  plot  in  the  middle  of  the 
 plot crossing each other, then, 
 Area of the path  =  ?? ( ?? + ?? - ?? )
 THREE DIMENSIONAL FIGURES 
 CUBE 
 All  sides  are  equal.  It  has  six  faces  and 
 12 edges. 
 ?  Volume =  ?? 
 3 
 ?  Total surface area  = 6  ?? 
 2 
 ?  Diagonal =  ??  3 
 ?  Sum of all edges = 12a 
 Here, a = length of the side 
 CUBOID 
 A  rectangular  body  having  3D  rectangular 
 shape, is called a cuboid. 
 ?  Volume  =  ??     ×     ??     ×     h 
 ?  Total surface area  = 
 2 ( ???? + ??h + ??h )
 ?  Diagonal =  ?? 
 2 
+ ?? 
 2 
+ h 
 2 
 BOX 
 A box has its shape like a cube or cuboid. 
 The amount that a box can hold or 
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 contain is called the capacity of the box. 
 Capacity means internal volume. 
 ?  Surface area of an open box 
 =  2  (  length  +  breadth  )  height ×
 + length  breadth    ×   
 =  2  ?? + ?? ( )    ×     h + ??  ×  ?? 
 ?  Capacity of box 
 =  ; ( ?? - 2  ?? )   ( ?? - 2  ?? )( h - 2  ?? )
 where, t = thickness of box 
 ?  Volume  of  the  material  of  the 
 box  =  External  volume  –  Internal 
 volume(  or  capacity)  = 
 ????h - ( ?? - 2  ?? )( ?? - 2  ?? )
( h - 2  ?? )
 ROOM 
 A  rectangular  room  has  four  walls 
 (surfaces)  and  opposite  walls  have  equal 
 area. 
 ?  Total Area of walls 
 =  2 ( ?? + ?? )    ×     h 
 ?  Total volume of the room 
 =  ??     ×     ??     ×     h 
 ?  Area of ?oor or roof =  ??     ×     ?? 
 CYLINDER 
 ?  Volume  of  cylinder  =  area  of 
 base  height = × p ?? 
 2 
 h 
 ?  Curved  surface  area  =  Perimeter 
 of base  height  = × 2p  ??h 
 ?  Total  surface  Area  =  curved 
 surface  area  +  area  of  both  the 
 circles 
 =  2p  ??h + 2p  ?? 
 2 
= 2p  ?? + h ( ) ?? 
 ?  When  the  rectangular  sheet  is 
 folded  along  its  length,  then  the 
 length  becomes  the 
 circumference  of  the  base  of 
 the  cylinder  and  breadth 
 becomes  the  height  of  the 
 cylinder. 
 HOLLOW CYLINDER 
 ?  Volume of hollow cylinder 
 = p( ?? 
 2 
- ?? 
 2 
) h 
 ?  Curved surface area 
 =  2p ( ?? + ?? ) h 
 ?  Total surface area 
 =  2p  ?? + ?? ( ) h + 2p ( ?? 
 2 
- ?? 
 2 
)
 =  2 p( ?? + ?? ) h + ?? - ?? { }
 Where,  R  =  External  radius  of 
 cylinder  ,  r  =  internal  radius  of 
 cylinder , h = height 
 PRISM 
 ?  Volume of prism 
 = area of base  height ×
 ?  Lateral  surface  area  =  Perimeter 
 of base  height ×
 ?  Total  surface  Area  =  Lateral 
 surface  area  +  area  of  base  and 
 top surface 
 CONE 
 A  solid  and  round  body  with  a  round  base 
 and pointed peak. 
 ?  Volume  =  × 
 1 
 3 
 ????????     ????????  × 
 height  = 
 1 
 3 
p ?? 
 2 
 h 
 ?  Slant height  (  ) =  ??  ?? 
 2 
+ h 
 2 
 ?  Curved surface area =  p  ???? 
= p  ??  ?? 
 2 
+ h 
 2 
 ?  Total  surface  area  =  p  ???? + p  ?? 
 2 
= p  ?? ( ?? + ?? )
 ?  Cone  formed  by  rotating  right 
 angled triangle about its height: 
 ?  Volume of cone so formed 
 = 
 1 
 3 
p ?? 
 2 
 ?? 
 ?  Similarly,  Cone  formed  by 
 rotating  right  angled  triangle 
 about its base: 
 Volume of cone so formed 
 = 
 1 
 3 
p ?? 
 2 
 ?? 
 ?  Similarly,  Cone  formed  by 
 rotating  right  angled  triangle 
 about its hypotenuse : 
 Volume of cone so formed 
 =  ,  (where  r  is  the  altitude 
 1 
 3 
p ?? 
 2 
 ?? 
 on hypotenuse and r =  ) 
 ??    ×    ?? 
 ?? 
 Note:  If  the  base  is  not  round,  it  will  be 
 called  a  pyramid.  A  pyramid  can  have 
 various  shapes  of  the  base  example: 
 square, rectangular, triangular etc. 
 PYRAMID 
 ?  Pyramid  means  a  structure  with 
 regular  polygon  as  its  base  and 
 sloping  sides  that  meet  in  a 
 point at the top. 
 ?  In  Pyramid,  with  n  sided  regular 
 polygon  at  its  base,  total 
 number of vertices = n + 1 
 ?  Volume =  ×base area × height 
 1 
 3 
 ?  Slant height  (l) =  ?? 
 2 
+ h 
 2 
 ?  Lateral surface area 
 = 
 ??????????????????     ×     ??????????     h??????h?? 
 2 
 ?  Total  Surface  area  =  Lateral 
 surface area + Area of base 
 TETRAHEDRON 
 It  is  a  3D  ?gure  made  by  joining  four 
 equilateral triangles. 
 ?  Volume  (V) 
 = 
 1 
 3 
    ×     ????????     ????????     ×     h??????h?? 
 V = ?   
 1 
 3 
×
 3 
 4 
 ?? 
 2 
×
 6 
 12 
 ?? +
 6 
 4 
 ?? 
( )
 V =  (Remember  this formula) ?
 2 
 12 
 ?? 
 3 
 ?  Lateral surface area =  a 
 2 
 3     × 
 3 
 4 
 ?  Total surface area 
 =  4 ×  a 
 2 
   
 3 
 4 
 ?  Height of Tetrahedron = 
 6 
 3 
 ?? 
 FRUSTUM OF CONE 
 If  a  cone  is  cut  by  a  plane  parallel  to  its 
 base,  so  as  to  divide  the  cone  into  two 
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 parts:  upper  part  and  lower  part,  then  the 
 lower part is called frustum. 
 ?  Slant height (l) 
 =  h 
 2 
+ ( ?? - ?? )
 2 
 ?  Curved Surface Area 
 = p ?? + ?? ( ) ?? 
 ?  Total surface area 
 = p ?? + ?? ( ) ?? + p  ?? 
 2 
+ p  ?? 
 2 
 =  p { ?? + ?? ( ) ?? + ?? 
 2 
+ ?? 
 2 
}
 ?  Volume  =  h    
 1 
 3 
 p ( ?? 
 2 
+ ?? 
 2 
+ ???? )
 SPHERE 
 ?  Volume of sphere  = 
 4 
 3 
p ?? 
 3 
 ?  Curved Surface area 
 = Total surface area =  4p  ?? 
 2 
 HOLLOW SPHERE OR 
 SPHERICAL SHELL: 
 ?  Volume of hollow sphere 
 = 
 4 
 3 
p( ?? 
 3 
- ?? 
 3 
)
 ?  Internal surface area  =  4p  ?? 
 2 
 ?  External surface area  =  4p  ?? 
 2 
 Here R = external radius  and 
 r = internal radius 
 HEMISPHERE 
 ?  Volume of the hemisphere 
 = 
 2 
 3 
   p    ?? 
 3 
 ?  Total surface area =  3p  ?? 
 2 
 ?  Curved surface area  =  2p  ?? 
 2 
 Where, r = radius 
 IMP. UNIT CONVERSION : 
 ?  1  = 1000 litres and  ?? 
 3 
 ?  1 litre = 1000  ???? 
 3 
 ?  1 meter = 10 decimeter 
 = 100 cm = 1000 millimeter 
 ?  1 meter =  decameter  1  0 
- 1 
 =  hectometer  1  0 
- 2 
 =  kilometer  1  0 
- 3 
 Variety Questions 
 Q.1.  The  base  of  a  right  prism  is  an 
 equilateral  triangle  with  each  side 
 measuring  4  cm.  If  the  lateral  surface 
 area  is  120  cm 
 2 
 ,  ?nd  the  volume  (in  cm 
 3 
 ) 
 of the prism. 
 SSC CHSL Tier II  (10/01/2024) 
 (a) 30  (b) 40  (c) 10  (d) 20  3  3  3  3 
 Q.2.  Some  ice  pieces,  spherical  in  shape, 
 of  diameter  6  cm  are  dropped  in  a 
 cylindrical  container  containing  some 
 juice  and  are  fully  submerged.  If  the 
 diameter  of  the  container  is  18  cm  and 
 level  of  juice  rises  by  40  cm,  then  how 
 many  ice  pieces  are  dropped  in  the 
 container ? 
 SSC CPO 05/10/2023 (3rd Shift) 
 (a) 90  (b) 80  (c) 85  (d) 95 
 Q.3.  Some  medicine  in  liquid  form  is 
 prepared  in  a  hemispherical  container  of 
 diameter  36  cm.  When  the  container  is 
 full  of  medicine,  the  medicine  is 
 transferred  to  small  cylindrical  bottles  of 
 diameter  6  cm  and  height  6  cm.  How 
 many  bottles  are  required  to  empty  the 
 container ? 
 SSC CPO 04/10/2023 (2nd Shift) 
 (a) 70  (b) 75  (c) 72  (d) 76 
 Q.4.  The  length  and  the  breadth  of  the 
 ?oor  of  a  rectangular  hall  are  126  feet 
 and  90  feet,  respectively.  What  will  be  the 
 area  (in  square  feet)  of  each  of  the 
 largest  identical  square  tiles  that  can  be 
 used  to  tile  this  ?oor  in  a  way  that  no  part 
 of the ?oor remains uncovered? 
 SSC CPO 03/10/2023 (3rd Shift) 
 (a) 196  (b) 256  (c) 324  (d) 484 
 Q.5.  The  radii  of  the  ends  of  a  frustum  of 
 a  solid  right-circular  cone  45  cm  high  are 
 28  cm  and  7  cm.  If  this  frustum  is  melted 
 and  reconstructed  into  a  solid  right 
 circular  cylinder  whose  radius  of  base 
 and  height  are  in  the  ratio  3  :  5,  ?nd  the 
 curved  surface  area  (in  cm 
 2 
 )  of  this 
 cylinder. [Use  .] p =
 22 
 7 
 SSC CPO 03/10/2023 (1st Shift) 
 (a) 4610  (b) 4620  (c) 4580  (d) 4640 
 Q.6.  A  square  and  a  rhombus  have  the 
 same  base  and  the  rhombus  is  inclined 
 at  45°,  then  what  will  be  the  ratio  of  the 
 area  of  the  square  to  the  area  of  the 
 rhombus? 
 SSC MTS 14/09/2023 (3rd Shift) 
 (a)  : 1  (b) 1 :  (c)  :  (d)  : 1  3     3  2     3  2    
 Q.7.  A  closed  wooden  box  measures 
 externally  10  cm  long.  8  cm  broad  and  6 
 cm  high.  If  the  thickness  of  the  wood  is 
 0.5  cm,  then  the  volume  of  wood 
 required is 
 SSC MTS 14/09/2023 (3rd Shift) 
 (a) 165 cm 
 3 
 (b) 300 cm 
 3 
 (c) 230 cm 
 3 
 (d) 150 cm 
 3 
 Q.8.  The  weight  of  a  cube  varies  directly 
 as  the  product  of  its  volume  and  its 
 density.  The  ratio  of  densities  of  the 
 materials  of  the  ?rst  cube  and  second 
 cube  is  27:16.  If  the  weight  of  the  ?rst 
 cube  is  4  times  the  weight  of  the  second 
 cube,  then  what  is  the  ratio  of  the  edges 
 of the ?rst cube and second cube? 
 SSC MTS 14/09/2023 (2nd Shift) 
 (a) 2 : 3  (b) 3 : 2  (c) 3 : 4  (d) 4 : 3 
 Q.9.  The  weight  of  a  circular  disc  varies 
 directly  as  the  product  of  the  square  of 
 the  radius  and  its  thickness.  Two 
 identical/similar  discs  have  their 
 thickness  in  the  ratio  of  16  :  9.  What  is 
 the  ratio  of  their  radii  if  the  weight  of  the 
 ?rst is four times that of the second ? 
 SSC MTS 12/09/2023 (3rd Shift) 
 (a) 3 : 2  (b) 2 : 3   (c) 1 : 2  (d) 2 : 1 
 Q.10.  A  path  of  uniform  width  2.5  m  runs 
 around  the  outside  of  a  rectangular  ?eld 
 of  dimensions  45  m  35  m.  What  is  the ×
 area of the path (in m 
 2 
 ) ? 
 SSC MTS 08/09/2023 (2nd Shift) 
 (a) 500  (b) 425  (c) 475  (d) 525 
 Q.11.  A  hemispherical  dome  of  a  building 
 needs  to  be  painted.  If  the  circumference 
 of  the  base  of  the  dome  is  154  cm,  then 
 ?nd  the  cost  of  painting  it  if  the  cost  of 
 painting is ?4 per 100 cm² (use  ). p =
 22 
 7 
 SSC CHSL 10/08/2023 (4th Shift) 
 (a)  ?150.92  (b)  ? 150.66 
 (c)  ? 105.29  (d)  ? 105.66 
 Q.12.  The  area  of  a  sector  of  a  circle  is 
 616  cm 
 2 
 with  a  central  angle  of  10°.  The 
 radius of the circle is _____.(use p =  ) 
 22    
 7    
 SSC CHSL 08/08/2023 (2nd Shift) 
 (a) 84 cm (b) 21 cm (c) 48 cm (d) 28 cm 
 Q.13.  Find  the  volume  of  the  largest  right 
 circular  cone  that  can  be  cut  out  from  a 
 cube  whose  edge  is  3  cm,  correct  to 
 1 
 2 
 two places of decimals (use p =  ). 
 22    
 7    
 SSC CHSL 07/08/2023 (4th Shift) 
 (a) 13.21 cm³  (b) 21.31 cm³ 
 (c) 11.23 cm³  (d) 12.13 cm³ 
 Q.14.  The  sum  of  the  radius  of  the  base 
 and  the  height  of  a  cylinder  is  42  m.  If  the 
 total  surface  area  of  the  cylinder  is  6336 
 ,  ?nd  the  curved  surface  area  of  the  ?? 
 2 
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Page 5


 Pinnacle  Day: 20th - 29th  Mensuration 
 Mensuration 
 2- D:  Under 2 dimensions we will study 
 ?  Triangles 
 ?  Quadrilateral 
 ?  Polygons 
 ?  Circle 
 3 - D: Under 3 dimensions we will study 
 ?  Cube                   
 ?  Cuboids 
 ?  Box                     
 ?  Cylinder 
 ?  Prism 
 ?  Cone 
 ?  Pyramid           
 ?  Frustum 
 ?  Sphere               
 ?  Hemisphere 
 ?  Tetrahedral 
 Note:  For  2D  ?gures,  area  and  perimeter 
 are  calculated  and  for  3D  ?gures,  volume 
 and surface area are calculated. 
 TWO DIMENSIONAL FIGURES 
 TRIANGLE 
 For a triangle with height, h, and base, b; 
 Area = 
 1 
 2 
    ×     ??     ×     h 
 Perimeter = Sum of all sides 
 Special cases: 
 1)  Equilateral  Triangle  -  All  sides  are 
 equal and each angle is  .  60° 
 ?  Area  = 
 3 
 4 
 ?? 
 2 
 ?  Height  = 
 3 
 2 
 ?? 
 ?  Perimeter =  3  ?? 
 ?  Inradius(r) = 
 ?? 
 2  3 
 ?  Circumradius(R) = 
 ?? 
 3 
 If  P1,  P2  and  P3  are  perpendicular  to  a 
 side from a point inside the triangle, then 
 ?  Height of triangle = P1+P2+P3 
 2)  Scalene  Triangle  –  All  sides  will  be 
 unequal lengths. 
 Area  =  ,  ?? ( ?? - ?? )( ?? - ?? )( ?? - ?? )
 This  formula  is  called  Heron's 
 formula. 
 Where, Semi-perimeter, 
 =  ?? 
 ??    +    ??    +    ?? 
 2 
 ?  Perimeter  =  ?? + ?? + ?? 
 3)  Isosceles  Triangle  :-  two  sides  and 
 two  angles  are  equal.  Altitude  bisects  the 
 base. 
 ?  Area  = 
 ?? 
 4 
 4  ?? 
 2 
- ?? 
 2 
 ?  Height =  ?? 
 2 
-
 ?? 
 2 
( )
 2 
 = 
 1 
 2 
 4  ?? 
 2 
- ?? 
 2 
 ?  Perimeter =  ?? + ?? + ?? 
= 2  ?? + ?? 
 4)  Right  angled  Triangle  :-  One  of  the 
 angles is 90  .  ° 
 Here, p = perpendicular , b = base and 
 h = hypotenuse 
 ?  Area = 
 1 
 2 
    ×     ??     ×     ?? 
 ?  Perimeter  =  ?? + ?? + h 
 ?  Pythagoras Theorem: 
 h 
 2 
= ?? 
 2 
+ ?? 
 2 
 ?  Inradius(r) = 
 ?? + ?? - h 
 2 
 ?  Circumradius(R) = 
 h 
 2 
 Note:  Common Pythagoras triplets: 
 (1,  1,  )  ;  (1,  2,  )  ;  (3,  4,  5);  (5,  12,  13);  2  5 
 (7, 24, 25); (20, 21, 29), (9, 40, 41); 
 (8, 15, 17) (12, 35, 37) (60, 11, 61) 
 (65, 72 97) (96, 110, 146). 
 Try remembering them. 
 QUADRILATERAL 
 A  ?gure  enclosed  by  four  sides  is  called 
 a  quadrilateral.  A  quadrilateral  has  four 
 angles  and  sum  of  these  angles  is  equal 
 to 360  .  ° 
 Special Cases: 
 1)  Parallelogram  -  It  is  a  quadrilateral 
 with opposite sides parallel and equal. 
 ?  Area  = base x height 
 ?  Perimeter = 2 (a + b) 
 ?  +  = 2(  +  )  ?? 
 1 
 2 
 ?? 
 2 
 2 
 ?? 
 2 
 ?? 
 2 
 ?  In  a  parallelogram,  opposite 
 sides  are  equal,  opposite  angles 
 are  equal  and  diagonals  bisect 
 each other. 
 2) Rhombus  - It is a parallelogram with 
 all four sides equal. The opposite angles 
 in a rhombus are equal. 
 Here, a = side;  and  are diagonals.  ?? 
 1 
 ?? 
 2 
 ?  Area = 
 1 
 2 
   × ?? 
 1 
× ?? 
 2 
 ?  Side (a) = 
 1 
 2 
 ?? 
 1 
 2 
+ ?? 
 2 
 2 
 ?  Perimeter = 4a 
 ?  4  ?? 
 2 
= ?? 
 1 
 2 
+ ?? 
 2 
 2 
 Diagonals bisect each other 
 at right angles . 
 3)  Trapezium  –  It  is  a  quadrilateral  with 
 one pair of opposite sides parallel. 
 Here,  a  and  b  are  parallel  sides  and  h  is 
 the  height  or  perpendicular  distance 
 between a and b. 
 ?  Area = 
 1 
 2 
 ×     h??????h?? 
× ??????  ????  ????????????????  ?????????? ( )
 = 
 1 
 2 
    ×     h        ×    ( ?? + ?? )
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 4)  Rectangle  – It is a parallelogram with 
 equal opposite sides and each angle is 90  ° 
 ?  Area = Length  Breadth = L  B × ×
 ?  Perimeter = 2 (L + B) 
 ?  Diagonal (d) =  ?? 
 2 
+ ?? 
 2 
 ?  Area  of  the  walls  of  a 
 rectangular room 
 =  2     ×    ( ?? + ?? )    ×     ?? 
 5)  Square  –  It  is  a  parallelogram  with  all 
 four  sides  equal  and  each  angle  is  equal 
 to 90  .  ° 
 ?  Area =  = ( ???????? )
 2 
= ?? 
 2 
   
 1 
 2 
 ?? 
 2 
 ?  Perimeter = 4  side = 4a ×
 ?  Diagonal (d) = a  2 
 NOTE:  Important  points  about 
 Quadrilaterals 
 1. The diagonals of a parallelogram 
 bisect each other. 
 2. Diagonal of a parallelogram divides it 
 into two triangles of equal area. 
 3. The diagonals of a rectangle are of 
 equal lengths and bisect each other. 
 4. The diagonals of a square are equal 
 and bisect each other at right angles. 
 5. A rhombus has unequal diagonals and 
 they bisect each other at right angles. 
 6. A parallelogram and a rectangle have 
 equal areas if they are on the same 
 base and between the same parallel 
 Lines. 
 REGULAR POLYGON 
 In  a  regular  polygon  all  sides  and  all 
 interior  angles  are  equal.  A  polygon  is 
 called  a  pentagon,  hexagon  ,  heptagon  , 
 octagon,  nonagon  and  a  decagon  as  they 
 have 5, 6, 7 , 8, 9, 10 sides, respectively. 
 If  each  side  of  a  regular  polygon  of  ‘n’ 
 sides is equal to ‘a’ then: 
 ?  Area of regular pentagon 
 = 
 6 . 9 
 4 
 ×  ?? 
 2 
 ?  Area of regular hexagon 
 =  6  × 
 3 
 4 
 ?? 
 2 
 ?  Each exterior angle  = 
 360° 
 ?? 
 ?  Each interior angle 
 =  180  Exterior angle  ° -   
 ?  Number of diagonals 
 = 
 ??  ??    -    3 ( )
 2 
 ?  Sum of all interior angles 
 = ( ?? - 2 ) ×180 
 ?  Sum of all exterior angle = 360° 
 CIRCLE 
 It  is  a  plane  ?gure  enclosed  by  a  line  on 
 which  every  point  is  equally  distant  from 
 a ?xed point (centre) inside the circle. 
 ?  Area = p  ?? 
 2 
 ?  Circumference  (perimeter ) 
 =  2p  ?? 
 ?  Diameter  = 2r 
 ?  Length of Arc  (AB) 
 =  2p  ??     ×    
?
 360° 
 ?  Area of sector AOB 
 = p ?? 
 2 
×
?
 360° 
 ?  Length  of  Arc  (AB)  =  ×  ?????????? 
 , (where angle in radian)  ???????????? 
 CIRCULAR RING 
 Here, R = radius of bigger ring , 
 r = radius of smaller ring 
 ?  Area =  p ( ?? 
 2 
- ?? 
 2 
)
 ?  Difference  in  circumference  of 
 both the rings =  2p  ?? - 2p  ?? 
 Short Cut methods/Tricks 
 1. If the length and breadth of a rectangle 
 are increased by a% and b%, the area 
 of the rectangle will be increased by 
 (  +  )%  ?? + ?? 
 ???? 
 100 
 2. If any of the two sides of the 
 rectangle is decreased then use ‘–ve’ 
 values  for that side. 
 3. All the sides of any two dimensional 
 ?gure changed by a%,  then its area will 
 change by (2  a  +  )%    
 ?? 
 2 
 100 
 Whenever there is a decrease, use 
 negative value for ‘a’ 
 4. If all the sides of any two dimensional 
 ?gure has changed (increased or 
 decreased) by a% then its perimeter 
 also changes by a%. In the case of a 
 circle such changes take place 
 because of the change in radius (or 
 diameter). 
 5. If the area of a square is ‘a’ square 
 unit. Then the area of the circle formed 
 with the same perimeter is given by 
 square units. 
 6. Area of a square inscribed in a 
 circle of radius ‘r’ is equal to  .  2  ?? 
 2 
 7. The area of the largest triangle 
 inscribed in a semicircle of radius r is 
 equal to  .  ?? 
 2 
 8. If a pathway of width x is made inside 
 or outside a rectangular plot of length 
 L and breath B, then area of the 
 pathway is 
 (i)  , if path is  2  ?? ( ?? + ?? + 2  ?? )
 made outside the plot 
 (ii)  , if a path is  2  ?? ( ?? + ?? - 2  ?? )
 made inside the plot. 
 9.  If  two  paths,  each  of  width  x  are  made 
 parallel  to  length  (L)  and  breadth  (B)  of 
 the  rectangular  plot  in  the  middle  of  the 
 plot crossing each other, then, 
 Area of the path  =  ?? ( ?? + ?? - ?? )
 THREE DIMENSIONAL FIGURES 
 CUBE 
 All  sides  are  equal.  It  has  six  faces  and 
 12 edges. 
 ?  Volume =  ?? 
 3 
 ?  Total surface area  = 6  ?? 
 2 
 ?  Diagonal =  ??  3 
 ?  Sum of all edges = 12a 
 Here, a = length of the side 
 CUBOID 
 A  rectangular  body  having  3D  rectangular 
 shape, is called a cuboid. 
 ?  Volume  =  ??     ×     ??     ×     h 
 ?  Total surface area  = 
 2 ( ???? + ??h + ??h )
 ?  Diagonal =  ?? 
 2 
+ ?? 
 2 
+ h 
 2 
 BOX 
 A box has its shape like a cube or cuboid. 
 The amount that a box can hold or 
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 contain is called the capacity of the box. 
 Capacity means internal volume. 
 ?  Surface area of an open box 
 =  2  (  length  +  breadth  )  height ×
 + length  breadth    ×   
 =  2  ?? + ?? ( )    ×     h + ??  ×  ?? 
 ?  Capacity of box 
 =  ; ( ?? - 2  ?? )   ( ?? - 2  ?? )( h - 2  ?? )
 where, t = thickness of box 
 ?  Volume  of  the  material  of  the 
 box  =  External  volume  –  Internal 
 volume(  or  capacity)  = 
 ????h - ( ?? - 2  ?? )( ?? - 2  ?? )
( h - 2  ?? )
 ROOM 
 A  rectangular  room  has  four  walls 
 (surfaces)  and  opposite  walls  have  equal 
 area. 
 ?  Total Area of walls 
 =  2 ( ?? + ?? )    ×     h 
 ?  Total volume of the room 
 =  ??     ×     ??     ×     h 
 ?  Area of ?oor or roof =  ??     ×     ?? 
 CYLINDER 
 ?  Volume  of  cylinder  =  area  of 
 base  height = × p ?? 
 2 
 h 
 ?  Curved  surface  area  =  Perimeter 
 of base  height  = × 2p  ??h 
 ?  Total  surface  Area  =  curved 
 surface  area  +  area  of  both  the 
 circles 
 =  2p  ??h + 2p  ?? 
 2 
= 2p  ?? + h ( ) ?? 
 ?  When  the  rectangular  sheet  is 
 folded  along  its  length,  then  the 
 length  becomes  the 
 circumference  of  the  base  of 
 the  cylinder  and  breadth 
 becomes  the  height  of  the 
 cylinder. 
 HOLLOW CYLINDER 
 ?  Volume of hollow cylinder 
 = p( ?? 
 2 
- ?? 
 2 
) h 
 ?  Curved surface area 
 =  2p ( ?? + ?? ) h 
 ?  Total surface area 
 =  2p  ?? + ?? ( ) h + 2p ( ?? 
 2 
- ?? 
 2 
)
 =  2 p( ?? + ?? ) h + ?? - ?? { }
 Where,  R  =  External  radius  of 
 cylinder  ,  r  =  internal  radius  of 
 cylinder , h = height 
 PRISM 
 ?  Volume of prism 
 = area of base  height ×
 ?  Lateral  surface  area  =  Perimeter 
 of base  height ×
 ?  Total  surface  Area  =  Lateral 
 surface  area  +  area  of  base  and 
 top surface 
 CONE 
 A  solid  and  round  body  with  a  round  base 
 and pointed peak. 
 ?  Volume  =  × 
 1 
 3 
 ????????     ????????  × 
 height  = 
 1 
 3 
p ?? 
 2 
 h 
 ?  Slant height  (  ) =  ??  ?? 
 2 
+ h 
 2 
 ?  Curved surface area =  p  ???? 
= p  ??  ?? 
 2 
+ h 
 2 
 ?  Total  surface  area  =  p  ???? + p  ?? 
 2 
= p  ?? ( ?? + ?? )
 ?  Cone  formed  by  rotating  right 
 angled triangle about its height: 
 ?  Volume of cone so formed 
 = 
 1 
 3 
p ?? 
 2 
 ?? 
 ?  Similarly,  Cone  formed  by 
 rotating  right  angled  triangle 
 about its base: 
 Volume of cone so formed 
 = 
 1 
 3 
p ?? 
 2 
 ?? 
 ?  Similarly,  Cone  formed  by 
 rotating  right  angled  triangle 
 about its hypotenuse : 
 Volume of cone so formed 
 =  ,  (where  r  is  the  altitude 
 1 
 3 
p ?? 
 2 
 ?? 
 on hypotenuse and r =  ) 
 ??    ×    ?? 
 ?? 
 Note:  If  the  base  is  not  round,  it  will  be 
 called  a  pyramid.  A  pyramid  can  have 
 various  shapes  of  the  base  example: 
 square, rectangular, triangular etc. 
 PYRAMID 
 ?  Pyramid  means  a  structure  with 
 regular  polygon  as  its  base  and 
 sloping  sides  that  meet  in  a 
 point at the top. 
 ?  In  Pyramid,  with  n  sided  regular 
 polygon  at  its  base,  total 
 number of vertices = n + 1 
 ?  Volume =  ×base area × height 
 1 
 3 
 ?  Slant height  (l) =  ?? 
 2 
+ h 
 2 
 ?  Lateral surface area 
 = 
 ??????????????????     ×     ??????????     h??????h?? 
 2 
 ?  Total  Surface  area  =  Lateral 
 surface area + Area of base 
 TETRAHEDRON 
 It  is  a  3D  ?gure  made  by  joining  four 
 equilateral triangles. 
 ?  Volume  (V) 
 = 
 1 
 3 
    ×     ????????     ????????     ×     h??????h?? 
 V = ?   
 1 
 3 
×
 3 
 4 
 ?? 
 2 
×
 6 
 12 
 ?? +
 6 
 4 
 ?? 
( )
 V =  (Remember  this formula) ?
 2 
 12 
 ?? 
 3 
 ?  Lateral surface area =  a 
 2 
 3     × 
 3 
 4 
 ?  Total surface area 
 =  4 ×  a 
 2 
   
 3 
 4 
 ?  Height of Tetrahedron = 
 6 
 3 
 ?? 
 FRUSTUM OF CONE 
 If  a  cone  is  cut  by  a  plane  parallel  to  its 
 base,  so  as  to  divide  the  cone  into  two 
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 parts:  upper  part  and  lower  part,  then  the 
 lower part is called frustum. 
 ?  Slant height (l) 
 =  h 
 2 
+ ( ?? - ?? )
 2 
 ?  Curved Surface Area 
 = p ?? + ?? ( ) ?? 
 ?  Total surface area 
 = p ?? + ?? ( ) ?? + p  ?? 
 2 
+ p  ?? 
 2 
 =  p { ?? + ?? ( ) ?? + ?? 
 2 
+ ?? 
 2 
}
 ?  Volume  =  h    
 1 
 3 
 p ( ?? 
 2 
+ ?? 
 2 
+ ???? )
 SPHERE 
 ?  Volume of sphere  = 
 4 
 3 
p ?? 
 3 
 ?  Curved Surface area 
 = Total surface area =  4p  ?? 
 2 
 HOLLOW SPHERE OR 
 SPHERICAL SHELL: 
 ?  Volume of hollow sphere 
 = 
 4 
 3 
p( ?? 
 3 
- ?? 
 3 
)
 ?  Internal surface area  =  4p  ?? 
 2 
 ?  External surface area  =  4p  ?? 
 2 
 Here R = external radius  and 
 r = internal radius 
 HEMISPHERE 
 ?  Volume of the hemisphere 
 = 
 2 
 3 
   p    ?? 
 3 
 ?  Total surface area =  3p  ?? 
 2 
 ?  Curved surface area  =  2p  ?? 
 2 
 Where, r = radius 
 IMP. UNIT CONVERSION : 
 ?  1  = 1000 litres and  ?? 
 3 
 ?  1 litre = 1000  ???? 
 3 
 ?  1 meter = 10 decimeter 
 = 100 cm = 1000 millimeter 
 ?  1 meter =  decameter  1  0 
- 1 
 =  hectometer  1  0 
- 2 
 =  kilometer  1  0 
- 3 
 Variety Questions 
 Q.1.  The  base  of  a  right  prism  is  an 
 equilateral  triangle  with  each  side 
 measuring  4  cm.  If  the  lateral  surface 
 area  is  120  cm 
 2 
 ,  ?nd  the  volume  (in  cm 
 3 
 ) 
 of the prism. 
 SSC CHSL Tier II  (10/01/2024) 
 (a) 30  (b) 40  (c) 10  (d) 20  3  3  3  3 
 Q.2.  Some  ice  pieces,  spherical  in  shape, 
 of  diameter  6  cm  are  dropped  in  a 
 cylindrical  container  containing  some 
 juice  and  are  fully  submerged.  If  the 
 diameter  of  the  container  is  18  cm  and 
 level  of  juice  rises  by  40  cm,  then  how 
 many  ice  pieces  are  dropped  in  the 
 container ? 
 SSC CPO 05/10/2023 (3rd Shift) 
 (a) 90  (b) 80  (c) 85  (d) 95 
 Q.3.  Some  medicine  in  liquid  form  is 
 prepared  in  a  hemispherical  container  of 
 diameter  36  cm.  When  the  container  is 
 full  of  medicine,  the  medicine  is 
 transferred  to  small  cylindrical  bottles  of 
 diameter  6  cm  and  height  6  cm.  How 
 many  bottles  are  required  to  empty  the 
 container ? 
 SSC CPO 04/10/2023 (2nd Shift) 
 (a) 70  (b) 75  (c) 72  (d) 76 
 Q.4.  The  length  and  the  breadth  of  the 
 ?oor  of  a  rectangular  hall  are  126  feet 
 and  90  feet,  respectively.  What  will  be  the 
 area  (in  square  feet)  of  each  of  the 
 largest  identical  square  tiles  that  can  be 
 used  to  tile  this  ?oor  in  a  way  that  no  part 
 of the ?oor remains uncovered? 
 SSC CPO 03/10/2023 (3rd Shift) 
 (a) 196  (b) 256  (c) 324  (d) 484 
 Q.5.  The  radii  of  the  ends  of  a  frustum  of 
 a  solid  right-circular  cone  45  cm  high  are 
 28  cm  and  7  cm.  If  this  frustum  is  melted 
 and  reconstructed  into  a  solid  right 
 circular  cylinder  whose  radius  of  base 
 and  height  are  in  the  ratio  3  :  5,  ?nd  the 
 curved  surface  area  (in  cm 
 2 
 )  of  this 
 cylinder. [Use  .] p =
 22 
 7 
 SSC CPO 03/10/2023 (1st Shift) 
 (a) 4610  (b) 4620  (c) 4580  (d) 4640 
 Q.6.  A  square  and  a  rhombus  have  the 
 same  base  and  the  rhombus  is  inclined 
 at  45°,  then  what  will  be  the  ratio  of  the 
 area  of  the  square  to  the  area  of  the 
 rhombus? 
 SSC MTS 14/09/2023 (3rd Shift) 
 (a)  : 1  (b) 1 :  (c)  :  (d)  : 1  3     3  2     3  2    
 Q.7.  A  closed  wooden  box  measures 
 externally  10  cm  long.  8  cm  broad  and  6 
 cm  high.  If  the  thickness  of  the  wood  is 
 0.5  cm,  then  the  volume  of  wood 
 required is 
 SSC MTS 14/09/2023 (3rd Shift) 
 (a) 165 cm 
 3 
 (b) 300 cm 
 3 
 (c) 230 cm 
 3 
 (d) 150 cm 
 3 
 Q.8.  The  weight  of  a  cube  varies  directly 
 as  the  product  of  its  volume  and  its 
 density.  The  ratio  of  densities  of  the 
 materials  of  the  ?rst  cube  and  second 
 cube  is  27:16.  If  the  weight  of  the  ?rst 
 cube  is  4  times  the  weight  of  the  second 
 cube,  then  what  is  the  ratio  of  the  edges 
 of the ?rst cube and second cube? 
 SSC MTS 14/09/2023 (2nd Shift) 
 (a) 2 : 3  (b) 3 : 2  (c) 3 : 4  (d) 4 : 3 
 Q.9.  The  weight  of  a  circular  disc  varies 
 directly  as  the  product  of  the  square  of 
 the  radius  and  its  thickness.  Two 
 identical/similar  discs  have  their 
 thickness  in  the  ratio  of  16  :  9.  What  is 
 the  ratio  of  their  radii  if  the  weight  of  the 
 ?rst is four times that of the second ? 
 SSC MTS 12/09/2023 (3rd Shift) 
 (a) 3 : 2  (b) 2 : 3   (c) 1 : 2  (d) 2 : 1 
 Q.10.  A  path  of  uniform  width  2.5  m  runs 
 around  the  outside  of  a  rectangular  ?eld 
 of  dimensions  45  m  35  m.  What  is  the ×
 area of the path (in m 
 2 
 ) ? 
 SSC MTS 08/09/2023 (2nd Shift) 
 (a) 500  (b) 425  (c) 475  (d) 525 
 Q.11.  A  hemispherical  dome  of  a  building 
 needs  to  be  painted.  If  the  circumference 
 of  the  base  of  the  dome  is  154  cm,  then 
 ?nd  the  cost  of  painting  it  if  the  cost  of 
 painting is ?4 per 100 cm² (use  ). p =
 22 
 7 
 SSC CHSL 10/08/2023 (4th Shift) 
 (a)  ?150.92  (b)  ? 150.66 
 (c)  ? 105.29  (d)  ? 105.66 
 Q.12.  The  area  of  a  sector  of  a  circle  is 
 616  cm 
 2 
 with  a  central  angle  of  10°.  The 
 radius of the circle is _____.(use p =  ) 
 22    
 7    
 SSC CHSL 08/08/2023 (2nd Shift) 
 (a) 84 cm (b) 21 cm (c) 48 cm (d) 28 cm 
 Q.13.  Find  the  volume  of  the  largest  right 
 circular  cone  that  can  be  cut  out  from  a 
 cube  whose  edge  is  3  cm,  correct  to 
 1 
 2 
 two places of decimals (use p =  ). 
 22    
 7    
 SSC CHSL 07/08/2023 (4th Shift) 
 (a) 13.21 cm³  (b) 21.31 cm³ 
 (c) 11.23 cm³  (d) 12.13 cm³ 
 Q.14.  The  sum  of  the  radius  of  the  base 
 and  the  height  of  a  cylinder  is  42  m.  If  the 
 total  surface  area  of  the  cylinder  is  6336 
 ,  ?nd  the  curved  surface  area  of  the  ?? 
 2 
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 Pinnacle  Day: 20th - 29th  Mensuration 
 cylinder  correct  to  two  places  of 
 decimals (use p =  ) 
 22    
 7    
 SSC CHSL 02/08/2023 (4th Shift) 
 (a) 2157.43  (b) 2571.43  ?? 
 2 
 ?? 
 2 
 (c) 2715.43  (d) 2517.43  ?? 
 2 
 ?? 
 2 
 Q.15.  Three  cubes  with  sides  in  the  ratio 
 of  3  :  4  :  5  are  melted  to  form  a  single 
 cube  whose  diagonal  is  18v3  cm.  The 
 sides of the three cubes are: 
 SSC CGL 20/07/2023 (1st shift) 
 (a) 21 cm, 28 cm and 35 cm 
 (b) 9 cm, 12 cm and 15 cm 
 (c) 18 cm, 24 cm and 30 cm 
 (d) 12 cm, 16 cm and 20 cm 
 Q.16.  The  perimeter  of  equilateral 
 triangle  is  3  ×  units.  Determine (
 4  16 
 3 
)
 the area of the triangle. 
 Matriculation Level 30/06/2023 (Shift - 4) 
 (a) 4 unit 
 2 
 (b) 1 unit 
 2 
 (c) 3 unit 
 2 
 (d) 2 unit 
 2 
 Q.17.  A  spherical  ball  of  diameter  8  cm 
 is  cut  into  two  equal  parts.  The  curved 
 area  of  one  such  part  has  to  be  painted 
 with  green  colour,  while  the  other  part 
 has  to  be  painted  with  red  colour.  The 
 bases  of  both  the  hemispheres  are  to  be 
 painted  with  blue  colour.  The  cost  of 
 painting  with  blue  is  ?2/cm²,  while  the 
 cost  of  painting  the  curved  area  is 
 ?3/cm².  What  will  be  the  cost  (in?)  of 
 painting the hemispheres? Take ? = 3.14 
 Matriculation Level 28/06/2023 (Shift - 4) 
 (a) ?451.92  (b) ?492.92 
 (c) ?803.84  (d) ?401.92 
 Q.18.  A  cube  of  side  5  cm,  has  been 
 assembled  using  125  cubes  each  of  1 
 cm  edge.  One  cube  is  removed  from  the 
 middle  of  each  of  the  faces  of  the  cube. 
 What  is  the  surface  area  (in  cm²)  of  the 
 remaining solid? 
 SSC CHSL Tier II (26/06/2023) 
 (a) 155  (b) 144  (c) 150  (d) 174 
 Q.19.  A  prism  and  a  pyramid  have  the 
 same  base  and  the  same  height.  Find  the 
 ratio  of  the  volumes  of  the  prism  and  the 
 pyramid. 
 SSC CGL Tier II (07/03/2023) 
 (a) 2 : 3  (b) 3 : 1  (c) 1 : 3    (d) 3 : 2 
 Q.20.  How  many  metres  of  2  m  wide 
 cloth  will  be  required  to  make  a  conical 
 tent  with  the  diameter  of  the  base  as  14 
 m  and  slant  height  as  9  m  to  ignore 
 wastage ? 
 SSC CGL 12/12/2022 (1st Shift) 
 (a) 66 m   (b) 88 m   (c) 99 m   (d) 77 m 
 Q.21.  The  minute  hand  of  a  clock  is  20 
 cm long. Find the area on the face of the 
 clock  swept  by  the  minute  hand  between 
 8 a.m. and 8:45 a.m. 
 SSC CGL 12/12/2022 (1st Shift) 
 (a)  cm 
 2 
 (b)  cm 
 2 
 6600 
 7 
 6600 
 9 
 (c)  cm 
 2 
 (d)  cm 
 2 
 6600 
 18 
 6600 
 14 
 Q.22.  The  external  diameter  of  an  iron 
 pipe  is  20  cm  and  its  length  is  12  cm.  If 
 the  thickness  of  the  pipe  is  1  cm,  ?nd  the 
 surface  area  of  the  pipe  (take  =  ) p
 22 
 7 
 correct to two places of decimal. 
 SSC CGL 06/12/2022 (2nd Shift) 
 (a) 1,662.67 cm 
 2 
 (b) 1,552.57 cm 
 2 
 (c) 1,442.48 cm 
 2 
 (d) 1,772.76 cm 
 2 
 Q.23.  If  the  length  of  certain  rectangle  is 
 decreased  by  4  cm  and  breadth  is 
 increased  by  2  cm,  it  would  result  in  a 
 square  of  the  same  area.  What  is  the 
 perimeter of the original rectangle ? 
 SSC CGL 06/12/2022 (1st Shift) 
 (a) 15 cm  (b) 24 cm  (c) 20 cm (d) 10 cm 
 Q.24.  Two  identical  circles  each  of  radius 
 30  cm  intersect  each  other  such  that  the 
 circumference  of  each  one  passes 
 through  the  center  of  the  other.  What  is 
 the area of the intersecting region? 
 SSC CGL 03/12/2022 (4th Shift) 
 (a) 400p - 250  cm²  3 
 (b) 300p - 150  cm²  3 
 (c) 500p - 350  cm²  3 
 (d) 600p - 450  cm²  3 
 Q.25.  The  area  of  a  trapezium  shaped 
 ?eld  is  1785  square  feet.  The  distance 
 between  the  two  parallel  sides  is  35  feet 
 and  one  of  the  parallel  sides  is  42  feet 
 long.  What  is  the  length  of  the  other 
 parallel side ? 
 SSC CPO 11/11/2022 (Evening) 
 (a) 60 ft  (b) 35 ft    (c) 40 ft    (d) 65 ft 
 Q.26.  If the sum of the diagonals of a 
 rhombus  is  L  and  the  perimeter  is  4P ,  ?nd 
 the area of the rhombus? 
 SSC CPO 11/11/2022 (Morning) 
 (a)  (L² - P²)  (b)  (L² - 4P²) 
 1 
 4 
 1 
 4 
 (c)  (L² - 4P²)  (d)  (L² + 3P²) 
 1 
 2 
 1 
 4 
 Q.27.  Water  is  ?owing  through  a 
 cylindrical  canal  with  an  internal 
 diameter  of  7  m  at  the  speed  of  18 
 km/h.Find  the  volume  of  water  ?owing 
 through the canal in 30 minutes. (take p
 =  ) 
 22 
 7 
 SSC CPO 09/11/2022 (Afternoon) 
 (a) 3,76,500 m³  (b) 3,56,500 m³ 
 (c) 3,66,500 m³  (d) 3,46,500 m³ 
 Q.28.  ABCDEF is a regular hexagon. Side 
 of the hexagon is 36 cm. What is the area 
 of the triangle ABC ? 
 SSC CGL Tier II (08/08/2022) 
 (a) 324  (b) 360  3  ???? 
 2 
 3  ???? 
 2 
 (c) 240  (d) 192  3  ???? 
 2 
 3  ???? 
 2 
 Q.29.  The  frustum  of  a  right  circular  cone 
 has  the  radius  of  the  base  as  5  cm, 
 radius  of  the  top  as  3  cm,  and  height  as  6 
 cm. What is its volume ? 
 SSC CHSL 27/05/2022 (Afternoon) 
 (a) 98 p  (b) 100 p  ???? 
 3 
 ???? 
 3 
 (c) 96 p  (d) 90 p  ???? 
 3 
 ???? 
 3 
 Q.30.  Three  sides  of  a  triangle  are 
 ,  and  ?? 
 2 
+ ?? 
 2 
( 2  ?? )
 2 
+ ?? 
 2 
 ?? 
 2 
+ ( 2  ?? )
 2 
 units.  What  is  the  area  (in  unit  squares) 
 of a triangle? 
 SSC CGL Tier II (03/02/2022) 
 (a)  (b) 3  (c) 4  (d) 
 5 
 2 
 ????  ????  ???? 
 3 
 2 
 ???? 
 Q.31.  A  square  park  has  been  divided 
 into  two  rectangles  of  equal  area.  If  the 
 perimeter  of  each  of  these  rectangles  is 
 39  m,  then  what  will  be  the  perimeter  of 
 the square park ? 
 SSC CHSL 09/08/2021 (Afternoon) 
 (a) 104 m  (b) 39 m  (c) 78 m  (d) 52 m 
 Q.32.  The  circumference  of  a  circle  is  ‘ap’ 
 units  and  the  area  of  the  circle  is  '  ?? p ' 
 square  units.  If  a:b  is  equal  to  4:5,  then 
 the radius of the circle is: 
 SSC CHSL 06/08/2021 (Evening) 
 (a) 3 cm  (b) 2.5 cm   (c) 5 cm   (d) 2 cm 
 Q.33.  The  area  of  a  triangular  ?eld  whose 
 sides  are  65m,  72m  and  97m  is  equal  to 
 the  area  of  a  rectangular  park  whose 
 sides  are  in  the  ratio  of  5  :  13.  What  is  the 
 perimeter (in m) of the rectangular park? 
 SSC CHSL 19/04/2021 (Morning) 
 (a) 108  (b) 180  (c) 216  (d) 144 
 Q.34.  A  bicycle  wheel  has  a  radius  of  42 
 cm.  It  makes  40  revolutions  in  25 
 seconds.  What  is  its  speed  (in  kmph,  up 
 to one decimal place) ? 
 SSC CHSL 16/04/2021 (Morning) 
 (a) 3.5  (b) 11.6  (c) 15.2  (d) 9.5 
 Q.35.  A  wire  is  bent  to  form  a  square  of 
 area  169  If  the  same  wire  is  bent  to  ???? 
 2 
.
 form  a  circle,  then  what  is  its  area  in  ???? 
 2 
 (to the nearest whole number) ? 
 SSC CHSL 13/04/2021 (Afternoon) 
 (a) 215  (b) 227  (c) 532  (d) 531 
 Q.36.  A  sector  of  radius  10.5  cm  with  the 
 central  angle  is  folded  to  form  a  120 
?
 cone  by  joining  the  two  bounding  radii  of 
 the sector. What is the volume (in  ) of  ???? 
 3 
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