Chapter 13. Surface Areas and Volumes

 Class 10 Maths Chapter 13. Surface Areas and Volumes Multiple Questions and Solution:

Question : The surface of a cube is  . The length of each edge is : [SEBA 2021]

(a) 15 m    (b) 10 m    (c) 8 m    (d) 5 m

Solution :  (b) 10 m              

[  let  be the length of the cube .

A/Q, 

 ]

Question : If the surface are of a cube is  . then its volume is : [SEBA 2023]

(a)     (b)     (c)    (d)

Solution :  (d)

[  let  be the length of the cube .

A/Q, 

The volume of the cube ]

Question : The volume and surface area of a sphere are equal , then the diameter of the sphere is : [SEBA 17]

(a)  3 units    (b)  6 units   (c) 2 units  (d) 4 units    

Solution:  (b)  6 units

[  We have ,  

 

   units 

units  ]

Question : Two identical cubes each of volume 64 are joined together end to end ,then the surface area of the resulting cuboid is :  [SEBA 2018 , 20]

  (a)           (b)           (c)        (d)

Solution: (d)

[ let  be the length of the cube .

   A/Q ,

 

For cuboid : Here ,

The surface area of cuboid

 

 

 

Question : The volume of a sphere is   . The radius of the sphere is : [SEBA 2019]

(a) 2 cm                    (c) 4 cm                (c) 6 cm               (d) 8

Solution:   (c)  6 cm

 [ let  be the radius of the sphere .

A/Q ,      

 

 

 

 

 cm       ]

Question : The volume of two spheres are in the ratio  then the ratio of their surface areas is : [SEBA16]

(a)  1 : 2                  (b) 2 : 3             (c) 9 : 16               (d) 16 : 9

Solution:   (d)  16 : 9

[ We have,

 

So,     ]   

Question :  A cuboid whose length , breadth and height are 15 cm , 10 cm and 20 cm respectively ,then its surface area is :

 (a)  1200      (b) 1400      (c)  1300     (d) 3000  

Solution:  (c)  1300   .

 [  Here,   cm , cm  and   cm

The surface area cuboid 

 

 

     ]

Question :  If two solid hemispheres of same base radius  are joined together along their bases, then  curved surface area of this new solid is :

 (a)      (b)       (c)       (d)

Solution:    (a)                    

Question : Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm . The diameter of each sphere is :

 (a) 4 cm             (b) 3 cm          (c) 2 cm          (d) 6 cm

Solution:  (c)  2 cm  

[ let  be the radius of the sphere .

Here ,  cm  and   cm

A/Q , 

 

  

 

  cm

 Therefore , the diameter  cm    ]  

Question : In the given figure, if  and  are radius and height of cylinder, then the total curve surface area of the cylinder is :

                

 (a)         (b)      (c)       (d)     

Solution:   (d)      .

 [ The total CSA of cylinder      ]

Question :  The number of solid spheres, each of diameter 6 cm than can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm is :    [CBSE 2014 ]

  (a)  3        (b)  5         (c)  4      (d)  6

Solution :   (b)  5

 [ Here, The radius of sphere  cm , the radius of cylinder  cm and  cm

The number of solid spheres  

        ]

Question : A rectangular sheet of paper 40 cm × 22 cm , is rolled to form a hollow cylinder of height 40 cm . The radius of the cylinder (in cm) is :   [CBSE 2014 F]

 (a)  3.5    (b)  7        (c)  8      (d)  5

Solution:    (a)  3.5

[ Here,    cm  ,   cm  and  cm

 Area of rectangular sheet of paper  

 A/Q , 

  

  cm     ]

  Class 10 Surface Areas and Volumes  Fill in the blank :

Question : If a sphere of the radius 7 cm , then the surface area of a sphere is   .

 Solution:   616 cm² .

 [ Here ,  cm 

  The surface area of  the sphere

          ]

Question : The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is    .

Solution:       .

[ Here ,  cm   and   cm

The volume of the largest right circular cone 

  

   

     ]  

  Class 10 Surface Areas and Volumes 2 Marks Questions and Solutions :

Question : A solid is in the shape of a cone standing on a hemisphere with both their radii being equal to 1 cm and the height of the cone is equal to its radius , then find the volume of the solid in terms of   .

Solution:   Here ,  cm

 The volume of the solid  

 

     .

Question : A metallic sphere of radius 4.2 cm is melted and recast into the shape of a cylinder of radius 6 cm . Find the height of the cylinder .

Solution:   let  be the height of the cylinder .

 Here ,  cm   and    cm

 A/Q ,       

  

   

 

    cm

Therefore, the height of the cylinder is 2.744 cm .

Question : Two identical cubes each of volume 125   are joined together end to end ,then find the surface area  of the resulting cuboid .

Solution:   Given ,      

 cm

Here ,  cm   ,  cm     and   

Therefore, the surface area of the resulting cuboid

 

  

           

Question : The surface area of a sphere is 616  . Find its radius .

Solution:   let ,  be the radius of the sphere .

 A/Q  ,   

  

  cm

  cm

 Therefore, the radius is 7 cm . 

Question : How many balls , each of radius 2 cm , can be made from a solid sphere of lead of radius 8 cm ?

Solution:  Here ,   cm   and  cm

 Therefore, number of  the balls 

   

Question : How many spherical bullets each of 6 cm in diameter can be cast from a rectangular block of metal  44 cm × 15 cm × 10 cm ?

Solution:   Here , Radius  cm ,  cm  ,  cm  and   cm

Therefore, the number of spherical bullets  

 

 

 

Question :  If the total surface area of a solid hemisphere is  , find its volume . [ Take  ]    [CBSE 2014]

Solution: let  be the radius of the hemisphere.

A/Q , 

 

 cm  

Therefore, the volume of hemisphere  

    

   

     

 Class 10 Surface Areas and Volumes 4 Marks Questions and Solutions   

Question :  A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter  of the hemisphere is equal to the edge of cube . Determine the surface area of the remaining solid . [SEBA 2023]

Solution: Here, diameter   and Radius

The surface area of the remaining solid = Area of cubical block + Area of hemisphere  – Area of circular top

 

 

Question :  A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. [SEBA 2021]

Solution : Here ,   , and

The volume of the toy = volume of the cone + volume of the hemisphere

Question :  A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius . The total height of the toy is 15.5 cm . Find the total surface area of the toy . [SEBA 2016 ,2019,2020]

Solution: For cone :  Radius , Height  cm

The slant height

 12.5 cm

The curve surface area of cone

 

   For hemisphere :

The curve surface area of hemisphere

The total surface area of the toy  

Question : From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out . Find the total surface area of the remaining solid to the nearest  . [SEBA 2017]

Solution:  Here , Diameter , Radius , Height  

 The slant height

 

 

 The total surface area of the remaining solid = C.S.A of cylinder + C.S.A. of cone + Area of circular top

   

 

 

 

  [appro.]

Question :  A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in given figure . If the height of the cylinder is 10 cm , and its base is of radius 3.5 cm , find the total surface area of the article . 

Solution: Here , Radius Height 

The total surface area of the article = C.S.A. of cylinder + Area of 2 hemisphere

 

 

 

   

Question :  A tent is in the shape of a cylinder surmounted by a conical top . If the height and diameter of the cylindrical part are 2.1 m and 4 m respectively, and the slant height of the top is 2.8 m, find the area of the canvas used for making the tent . Also, find the cost of the canvas of the tent at the rate of Rs 500 per  . (Note that the base of the tent will not be covered with canvas)

Solution: Here , Height  , Diameter , Radius and the slant height

The area of the canvas of the tent = Area of the cylinder + C.S. area of cone  

 

 

The cost of the canvas of the tent  .

Question :  A gulab jamun, contains sugar syrup up to about 30% of its volume . Find approximately how much syrup would be found in 45 gulab jamuns , each shaped liked a cylinder with two hemispherical ends with length 5 cm and diameter 2.8 cm (see Fig. 13.15)

     

Solution:  Here,   , and

The volume of a gulab jamuns = The volume of the hemisphere + The volume of the cylinder + The volume of the hemisphere

The volume of 45 gulap jamuns

The volume of gulap jamun contains in sugar syrup

Question : A pen stand made of wood is in the shape of a cuboid with four conical depressions to hold pens . The dimensions of the cuboid are 15cm by 10cm by 3.5 cm . The radius of each of the depressions is 0.5 cm and the depth is 1.4 cm . Find the volume of wood in the entire stand .

     

Solution: For cuboid : Here , length  cm , breadth  cm , height  cm

The volume of a cuboidal wood

 

For cone : Here , and

 The volume of a  cone

The volume of  4 cone

Therefore, the volume of wood in the entire stand

  

Question :  A vessel is in the form of an inverted cone . Its height is 8 cm and the radius of its top , which is open, is 5 cm . It is filled with water up to the brim . When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel , one-fourth of the water flows out . Find the number of lead shots dropped in the vessel .

Solution:  For cone : Here, Height of cone  cm  and Radius  cm

  The volume of cone

  For Sphere :  Here,  Radius  cm 

  The volume of the sphere

 The volume of the water that flows out of the cone

(the volume of the cone)                

 Therefore, the number of lead shots

Question :  A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm , which is surmounted by another cylinder of height 60 cm and radius 8 cm . Find the mass of the pole , given that 1 cm³ of iron has approximately 8g mass . (Use )

Solution :  For big cylinder : Here ,  Diameter  , Radius and Height

The volume of the big cylindrical pole

 

 

For small cylinder : Here ,

 Radius , Height

The volume of the big cylindrical pole

 

 

The volume of solid pole

The mass of the pole