Class 10 Maths · Chapter 12

Surface Areas and Volumes

90 Class 10 Maths questions on Surface Areas and Volumes from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 3 of 5

1. Q41 Exemplar · Exercise 12.2
   A solid ball is ​exactly fitted inside the cubical box of side $a$. ​The volume of the ​ball ​is $\frac{4}{3}\pi a^3$. Write 'True' ​or ​'False' and justify ​your answer.
   Show answer

   False. The ball's radius is $a/2$, so its volume is $\frac{4}{3}\pi (a/2{)}^3=\frac{\pi a^3}{6}$, not $\frac{4}{3}\pi a^3$.

   Step-by-step solution →
2. Q42 Exemplar · Exercise 12.2
   State ​whether the following statement is true or false and justify: ​The volume of the frustum of a ​cone ​is $\frac{1}{3}\pi h[r_1^2+r_2^2-r_1{r}_2]$, where $h$ is the ​vertical height of ​the ​frustum and $r_1,r_2$ are the radii of ​the ends.
   Show answer

   False

   Step-by-step solution →
3. Q43 Exemplar · Exercise 12.2
   The capacity of a cylindrical vessel with ​a hemispherical ​portion raised ​upward ​at the ​bottom ​as ​shown in the Fig. 12.7 ​is $\frac{\pi r^2}{3}[3h-2r]$.

   Show answer

   Capacity $=\frac{\pi r^2}{3}(3h-2r)$, which is the required result.

   Step-by-step solution →
4. Q44 Exemplar · Exercise 12.2
   The curved surface area of ​a frustum ​of ​a cone is $\pi l(r_1+r_2)$, where $l=\sqrt{h^2+(r_1+r_2{)}^2}$, $r_1$ and $r_2$ are the ​radii of ​the ​two ends of the frustum and $h$ is the ​vertical height.
   Show answer

   False

   Step-by-step solution →
5. Q45 Exemplar · Exercise 12.2
   An open metallic ​bucket is in the shape of a frustum of a cone, mounted on a hollow ​cylindrical base made of ​the same metallic sheet. The surface area ​of ​the metallic sheet ​used ​is equal to curved surface area of frustum of a cone + area of circular ​base + curved surface area of cylinder.
   Show answer

   True

   Step-by-step solution →
6. Q46 Exemplar · Exercise 12.3
   Three metallic ​solid cubes whose edges ​are 3 cm, 4 cm ​and 5 cm are ​melted ​and formed into ​a single cube. Find the ​edge of the cube so formed.
   Show answer

   The edge of the cube formed is $6$ cm.

   Step-by-step solution →
7. Q47 Exemplar · Exercise 12.3
   How many ​shots ​each having diameter 3 cm can be ​made from ​a ​cuboidal lead ​solid ​of ​dimensions $9\text{ cm}\times 11\text{ cm}\times 12\text{ cm}$?
   Show answer

   $84$ spherical lead shots can be made.

   Step-by-step solution →
8. Q48 Exemplar · Exercise 12.3
   A ​bucket is in the form of a ​frustum of a cone and holds $28.490$ litres ​of ​water. The radii of the ​top and bottom are $28$ cm and $21$ cm, ​respectively. ​Find the height of the bucket.
   Show answer

   The height of the bucket is $15$ cm.

   Step-by-step solution →
9. Q49 Exemplar · Exercise 12.3
   A cone of radius 8 cm and height 12 cm is ​divided ​into two parts by a ​plane ​through the mid-point of its axis ​parallel ​to its base. Find the ratio ​of the volumes of two ​parts.
   Show answer

   The ratio of the volumes (small cone : frustum) is $1:7$.

   Step-by-step solution →
10. Q50 Exemplar · Exercise 12.3
    Two identical cubes each of volume $64{\text{cm}}^3$ are joined ​together end to end. ​What ​is ​the ​surface ​area of the ​resulting cuboid?
    Show answer

    The surface area of the resulting cuboid is $160{\text{cm}}^2$.

    Step-by-step solution →
11. Q51 Exemplar · Exercise 12.3
    From a solid cube ​of ​side 7 cm, ​a conical cavity of height ​7 ​cm ​and ​radius 3 cm is hollowed out. Find ​the volume of the remaining solid.
    Show answer

    Volume of the remaining solid $=277$ cm${}^3$.

    Step-by-step solution →
12. Q52 Exemplar · Exercise 12.3
    Two ​cones with same base ​radius 8 cm and height 15 cm are joined together along ​their bases. ​Find ​the surface area ​of the shape so formed.
    Show answer

    Surface area $=\frac{5984}{7}\approx 854.86{\text{cm}}^2$.

    Step-by-step solution →
13. Q53 Exemplar · Exercise 12.3
    Two solid cones A and B are placed in a cylindrical ​tube as ​shown in Fig. 12.9. The ​ratio ​of their capacities ​is 2:1. Find the heights ​and ​capacities of the ​cones. Also, find the volume of the remaining portion of the cylinder. (The cylindrical tube has length 21 cm and diameter 6 cm.)

    Show answer

    Heights: $h_A=14$ cm, $h_B=7$ cm; capacities $V_A=42\pi \approx 131.9$ cm${}^3$, $V_B=21\pi \approx 66.0$ cm${}^3$; remaining volume $=126\pi \approx 395.8$ cm${}^3$.

    Step-by-step solution →
14. Q54 Exemplar · Exercise 12.3
    An ice cream cone full of ice ​cream having radius ​5 cm and height 10 cm ​as shown in ​Fig. 12.10. Calculate the volume of ice cream, ​provided that its $\frac{1}{6}$ ​part is left unfilled ​with ice ​cream.

    Show answer

    Volume of ice cream $=\frac{625}{6}\pi \approx 327.4{\text{cm}}^3$ (taking $\pi =\frac{22}{7}$).

    Step-by-step solution →
15. Q55 Exemplar · Exercise 12.3
    Marbles of diameter 1.4 ​cm are ​dropped into a cylindrical beaker of diameter 7 cm containing some water. Find the number ​of marbles ​that should be ​dropped into the ​beaker so ​that the water ​level rises by 5.6 cm.
    Show answer

    $150$ marbles should be dropped into the beaker.

    Step-by-step solution →
16. Q56 Exemplar · Exercise 12.3
    How ​many spherical lead ​shots each of diameter 4.2 cm can be obtained from a solid ​rectangular lead ​piece with ​dimensions 66 cm, 42 ​cm and 21 cm.
    Show answer

    $1500$ spherical lead shots can be obtained.

    Step-by-step solution →
17. Q57 Exemplar · Exercise 12.3
    How many spherical lead shots of diameter 4 cm can be made out of ​a solid cube ​of ​lead whose ​edge ​measures ​44 ​cm?
    Show answer

    $N=2541$ spherical lead shots can be made.

    Step-by-step solution →
18. Q58 Exemplar · Exercise 12.3
    A wall 24 ​m long, 0.4 m thick and 6 ​m high is constructed with the bricks each ​of dimensions $25\text{ cm}\times 16\text{ cm}\times 10\text{ cm}$. If the mortar occupies $\frac{1}{10}$th of ​the volume of the wall, then ​find the number ​of bricks ​used in constructing the wall.
    Show answer

    $12960$ bricks were used in constructing the wall.

    Step-by-step solution →
19. Q59 Exemplar · Exercise 12.3
    Find ​the number of metallic circular discs with 1.5 cm ​base ​diameter and of height 0.2 cm to ​be melted to form a ​right ​circular cylinder of height 10 cm and diameter ​4.5 cm.
    Show answer

    $n=450$ discs must be melted.

    Step-by-step solution →
20. Q60 CBSE 2026
    Two wooden solid ​hemispheres of ​same radii $r$, are ​joined at a point, as shown in the figure. The total ​surface area of the object ​is:

    1. (a)$4\pi r^2$
    2. (b)$6\pi r^2$
    3. (c)$3\pi r^2$
    4. (d)$5\pi r^2$
    Show answer

    (b) $6\pi r^2$

    Step-by-step solution →