21. Chapter 21. Surface Areas and Volumes of Solid Figures NIOS Class 10 Mathematics Textbook Solutions
Chapter 21. Surface Areas and Volumes of Solid Figures NIOS Class 10 Mathematics Textbook Solutions for Exam Preparation
Chapter 21. SURFACE AREAS AND VOLUMES OF SOLID FIGURES
CHECK YOUR PROGRESS 21.1
1. Find the surface area and volume of a cuboid of length 6m, breadth 3m and height 2.5m.
Solution: Here, 
The surface area of a cuboid 
m²
m²
The volume of a cuboid 
m³
2. Find the surface area and volume of a cube of edge 3.6 cm.
Solution: Here, 
cm
The surface area of a cube 
The volume of a cube 
3. Find the edge of a cube whose volume is 3375 cm³ . Also, find its surface area.
Solution: Let , a the edge of a cube .
A/Q, 
cm
The surface area of a cube 
4. The external dimensions of a closed wooden box are 42 cm × 32 cm × 27 cm. Find the internal volume of the box, if the thickness of the wood is 1cm.
Solution: Here, 
,
and 
The internal volume of the box
5. The length, breadth and height of a godown are 12m, 8m and 6 metres respectively. How many boxes it can hold if each box occupies 1.5 m³ space?
Solution: Here, l=12 m , b=8 m , h=6 m
The volume of a godown =l×b×h=12×8×6=576 m³
The volume of a box = 1.5 m³
The number of the boxes 
6. Find the length and surface area of a wooden plank of width 3m, thickness 75 cm and volume 33.75 m³.
Solution: Let , 
be the length of a wooden plank .
Here,
A/Q, 
The surface area of a wooden plank 
7. Three cubes of edge 8 cm each are joined end to end to form a cuboid. Find the surface area and volume of the cuboid so formed.
Solution: Here, l=8+8+8=24 cm , b=8 , h=8 cm
The surface area of the cuboid
=2×448=896 cm²
The volume of the cuboid 
8. A room is 6m long, 5m wide and 4m high. The doors and windows in the room occupy 4 square metres of space. Find the cost of papering the remaining portion of the walls with paper 75cm wide at the rate of Rs 2.40 per metre.
Solution: Here, l=6 m , b=5 m , h=4 m
The area of four walls 
The area of the four walls excluding the doors and windows = 88 − 4 = 84 m².
The length of the paper on wall 
The cost of papering the remaining portion of the walls =Rs 112×2.40=Rs 268.80
9. Find the length of the longest rod that can be put in a room of dimensions 6m × 4m ×3m.
Solution: Here, l=6m , b=4m , h=3m
The length of the longest rod 
CHECK YOUR PROGRESS 21.2
1. Find the curved surface area, total surface area and volume of a right circular cylinder of radius 5 m and height 1.4 m.
Solution: Here, r=5 m , h=1.4 m
The curved surface area of a right circular cylinder 
The total surface area of a right circular cylinder 
The volume of a right circular cylinder 
2. Volume of a right circular cylinder is 3080 cm³ and radius of its base is 7 cm. Find the curved surface area of the cylinder.
Solution: Here, r=7 cm
Let, h be the height of a cylinder .
A/Q, 
The curved surface area of the cylinder 
3. A cylindrical water tank is of base diameter 7 m and height 2.1 m. Find the capacity of the tank in litres.
Solution: Here,
The capacity of the cylindrical water tank 
4. Length and breadth of a paper is 33 cm and 16 cm respectively. It is folded about its breadth to form a cylinder. Find the volume of the cylinder.
Solution: Let, r be the radius of a cylinder .
Here, 
The perimeter of a cylindrical paper = 33 cm
The volume of the cylinder
5. A cylindrical bucket of base diameter 28 cm and height 12 cm is full of water. This water is poured in to a rectangular tub of length 66 cm and breadth 28 cm. Find the height to which water will rise in the tub.
Solution: For a cylindrical bucket : 
The volume of a cylindrical bucket 
For a rectangular tub: Here, 
Let h be the height to which the water will rise in the tub.
The volume of the rectangular tub 
A/Q, 
6. A hollow metallic cylinder is open at both the ends and is of length 8 cm. If the thickness of the metal is 2 cm and external diameter of the cylinder is 10 cm, find the whole curved surface area of the cylinder (use π = 3.14). [Hint: whole curved surface = Internal curved surface + External curved surface]
Solution: Here, 
The internal curved surface of the cylinder 
Outer cylinder : 
The external curved surface of the cylinder 
The whole curved surface area of the cylinder = Internal curved surface + External curved surface
= 150.86 + 251.43 =402.29 cm² (approx.)
CHECK YOUR PROGRESS 21.3
1. Find the curved surface area, total surface area and volume of a right circular cone whose base radius and height are respectively 5 cm and 12 cm.
Solution: Here, r=5 cm , h=12 cm , 
The curved surface area of a right circular cone =πrl 
The total surface area of a right circular cone 
The volume of a right circular cone 
2. Find the volume of a right circular cone of base area 616 cm² and height 9 cm.
Solution: Here, h=9 cm
We have, 
The volume of a right circular cone
3. Volume of a right circular cone of height 10.5 cm is 176 cm³ . Find the radius of the cone.
Solution: Here, h=10.5 cm
Let, r be the radius of the cone .
A/Q, 
Therefore, the radius of the cone is 4 cm .
4. Find the length of the 3 m wide canvas required to make a conical tent of base radius 9 m and height 12 m (use π = 3.14).
Solution: Here, 
The curve surface area of the canvas 
Therefore, the length of the canvas required to make a conical tent 
5. Find the curved surface area of a right circular cone of volume 12936 cm³ and base diameter 42 cm.
Solution: Here, 
Let, h be the height of a right circular cone .
A/Q, 
cm 
The slant height 
The curved surface area of a right circular cone 
CHECK YOUR PROGRESS 21.4
1. Find the surface area and volume of a sphere of radius 14 cm.
Solution: Here, r=14 cm
The surface area of a sphere 
=
The volume of a sphere 
2. Volume of a sphere is 38808 cm³ . Find its radius and hence its surface area.
Solution: Let, r be the radius of a sphere .
A/Q, 
The surface area of a sphere 
3. Diameter of a hemispherical toy is 56 cm. Find its
(i) curved surface area (ii) total surface area (iii) volume
Solution: Here, 
(i) The curved surface area of a hemispherical toy 
(ii) The total surface area of a hemispherical toy 
(iii) The volume of a hemispherical toy 
4. A metallic solid ball of radius 28 cm is melted and converted into small solid balls of radius 7 cm each. Find the number of small balls so formed.
Solution: Here, 
Therefore, the number of small balls 
TERMINAL EXERCISE
1. Fill in the blanks:
(i) Surface area of a cuboid of length l, breadth b and height h = _________
(ii) Diagonal of the cuboid of length l, breadth b and height h = ___________
(iii) Volume of the cube of side a = __________.
(iv) Surface area of cylinder open at one end = ______, where r and h are the radius and height of the cylinder.
(v) Volume of the cylinder of radius r and height h = __________
(vi) Curved surface area of cone = ____________, where r and l are respectively the ______ and _________ of the cone.
(vii) Surface area of a sphere of radius r = __________
(viii) Volume of a hemisphere of radius r = ___________
Answer: (i) Surface area of a cuboid of length l, breadth b and height h 
(ii) Diagonal of the cuboid of length l, breadth b and height 
(iii) Volume of the cube of side a 
(iv) Surface area of cylinder open at one end 
, where r and h are the radius and height of the cylinder.
(v) Volume of the cylinder of radius r and height h 
.
(vi) Curved surface area of cone , where r and l are respectively the radius and slant height of the cone.
(vii) Surface area of a sphere of radius r 
(viii) Volume of a hemisphere of radius 
2. Choose the correct answer from the given four options:
(i) The edge of a cube whose volume is equal to the volume of a cuboid of dimensions 63 cm × 56 cm × 21 cm is
(A) 21 cm (B) 28 cm (C) 36 cm (D) 42 cm
Answer: (D) 42 cm .
[ Let, a be the edge of a cube .
Here, 
A/Q, The volume of a cube = The volume of a cuboid
]
(ii) If radius of a sphere is doubled, then its volume will become how many times of the original volume?
(A) 2 times (B) 3 times (C) 4 times (D) 8 times
Answer: (D) 8 times .
[ Here, 
The volume of a sphere 
]
(iii) Volume of a cylinder of the same base radius and the same height as that of a cone is
(A) the same as that of the cone (B) 2 times the volume of the cone
(C) 1/3 times the volume of the cone (D) 3 times the volume of the cone.
Answer: (D) 3 times the volume of the cone.
[ The volume of a cone = 
And the volume of a cylinder 
]
3. If the surface area of a cube is 96 cm² , then find its volume.
Solution: Let, 
be the side length of a cube .
A/Q, 
The volume of a cube 
4. Find the surface area and volume of a cuboid of length 3m, breadth 2.5 m and height 1.5 m.
Solution: Here, 
The surface area of a cuboid 
The volume of a cuboid 
5. Find the surface area and volume of a cube of edge 1.6 cm.
Solution: Here, 
The surface area of a cube 
The volume of a cube 
6. Find the length of the diagonal of a cuboid of dimensions 6 cm × 8 cm × 10 cm.
Solution: Here, 
The length of the diagonal of a cuboid
7. Find the length of the diagonal of a cube of edge 8 cm.
Solution: Here, 
The length of the diagonal 
8. Areas of the three adjecent faces of cuboid are A, B and C square units respectively and its volume is V cubic units. Prove that V² = ABC.
Solution: Let l , b and h be the dimensions of the cuboid length, breadth and height respectively .
We have, 
, 
, 
The volume of the cubiod 
So, 
9. Find the total surface area of a hollow cylindrical pipe open at the ends if its height is 10 cm, external diameter 10 cm and thickness 0.1 cm (use π = 3.14).
Solution: Here, 
, thickness = 0.1 cm
The total surface area 
10. Find the slant height of a cone whose volume is 12936 cm³ and radius of the base is 21 cm. Also, find its total surface area.
Solution: Let , h be the height of a cone .
Here, 
A/Q,
The slant height 
The total curve surface area of a cone
11. A well of radius 5.6 m and depth 20 m is dug in a rectangular field of dimensions 150 m × 70 m and the earth dug out from it is evenly spread on the remaining part of the field. Find the height by which the field is raised.
Solution: Here, 
The volume of a well (cylinder) 
The area of the rectangular field 
Area of well top 
The area of the remaining part of the field 
Let h be the rise in height .
A/Q, 
Therefore, the rise in height is 18.95 cm .
12. Find the radius and surface area of a sphere whose volume is 38808 m³ .
Solution: Let , r be the radius of a sphere .
A/Q, 
The surface area of a sphere 
13. In a room of length 12 m, breadth 4 m and height 3 m, there are two windows of dimensions 2m × 1 m and a door of dimensions 2.5 m × 2 m. Find the cost of papering the walls at the rate of Rs 30 per m².
Solution: Here, 
The lateral surface area of a cuboid (= Area of walls)
The paperable area of the room = Area of walls – 2 × Area of window – Area of a door
Therefore, the cost of papering the walls 
14. A cubic centimetre gold is drawn into a wire of diameter 0.2 mm. Find the length of the wire. (use π = 3.14).
Solution: Here, 
A/Q, 
Therefore, the length of the wire is 31.85 m .
15. If the radius of a sphere is tripled, find the ratio of the
(i) Volume of the original sphere to that of the new sphere.
(ii) surface area of the original sphere to that of the new sphere.
Solution: Here, 
(i) We have, 
(ii) We have, 
16. A cone, a cylinder and a hemisphere are of the same base and same height. Find the ratio of their volumes.
Solution: Here, 
The volume of a cone, a cylinder and a hemisphere 
17. Slant height and radius of the base of a right circular cone are 25 cm and 7 cm respectively. Find its
(i) curved surface area (ii) total surface area, and (iii) volume
Solution: Here, 
(i)The curved surface area of a right circular cone 
(ii) The total surface area of a right circular cone
(iii) The volume of a right circular cone 
18. Four cubes each of side 5 cm are joined end to end in a row. Find the surface and the volume of the resulting cuboid.
Solution: Here, 
The surface of the resulting cuboid 
The volume of the resulting cuboid 
19. The radii of two cylinders are in the ratio 3 : 2 and their heights are in the ratio 7 : 4. Find the ratio of their
(i) volumes. (ii) curved surface areas.
Solution: Here, 
(i) We have , 
The ratio of the volumes of the two cylinders is 63 : 16 .
(ii) We have, 
The ratio of the curved surface areas of the two cylinders is 21 : 8 .
20. State which of the following statements are true and which are false:
(i) Surface area of a cube of side a is 6a².
(ii) Total surface area of a cone is πrl, where r and l are respectively the base radius and slant height of the cone.
(iii) If the base radius and height of cone and hemisphere are the same, then volume of the hemisphere is thrice the volume of the cone.
(iv) Length of the longest rod that can be put in a room of length l, breadth b and height h is 
(v) Surface area of a hemisphere of radius r is 2πr².
Answer: (i) Surface area of a cube of side a is 6a². → True
[A cube has 6 square faces, each of area a².
Total surface area 
.]
(ii) Total surface area of a cone is πrl, where r and l are respectively the base radius and slant height of the cone. → False
[ 
is the curved surface area of a cone, not the total surface area.]
(iii) If the base radius and height of cone and hemisphere are the same, then volume of the hemisphere is thrice the volume of the cone. → False
[ We have, 
So hemisphere is twice the cone, not thrice. ]
(iv) Length of the longest rod that can be put in a room of length l, breadth b and height h is 
.→ True
[ The longest rod that fits in a room is the space diagonal of the cuboid, given by 
.]
(v) Surface area of a hemisphere of radius r is 2πr². → False
[ 2πr² is the curved surface area of a hemisphere.
The total surface area 
]
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