Surface Areas & Volumes (Exercise 12.1) NCERT Solutions - Mathematics (Maths) Class 10

Q1. 2 cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
Sol.  Volume of each cube = 64 cm³
∴ Total volume of the two cubes = 2 × 64 cm³ = 128 cm³
Let the edge of each cube = x
∴ x³ = 64 = 4³
⇒ x =4 cm
⇒ Now, the Length of the resulting cuboid 'l' = 2x cm = 8cm
⇒ Breadth of the resulting cuboid 'b' = x cm = 4 cm
⇒ Height of the resulting cuboid 'h' = x cm = 4 cm
∴ Surface area of the cuboid = 2 (lb + bh + hl)
= 2 [(8 × 4) + (4 × 4) + (4 × 8)]
= 2 [32 + 16 + 32] cm² = 2 [80] cm² = 160 cm².

Q2. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Sol.  For the cylindrical part
 Radius (r) = 7 cm
 Height (h) = 6 cm
∴ Curved surface area
= 2πrh

For hemispherical part
  Radius (r) = 7 cm
∴ Surface area = 2πr²

∴ Total surface area = CSA of cylinder + CSA of hemisphere
= (264 + 308) cm² = 572 cm².

Q3. A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of the same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy.
Sol. Here, r = 3.5 cm
∴ h = (15.5 − 3.5) cm = 12.0 cm
⇒ Surface area of the conical part = πrl
⇒ Surface area of the hemispherical part = 2πr²

Q4. A cubical block of side 7 cm is surmounted by a hemisphere. What is the greatest diameter the hemisphere can have? Find the surface area of the solid.
Sol. Side of the block = 7 cm
⇒ The greatest diameter of the hemisphere = 7 cm
Surface area of the solid:
= [Total S.A. of the cubical block] + [S.A. of the hemisphere] − [Base area of the hemisphere]
= (6 × l²) + 2πr² − πr²

Q5. A hemispherical depression is cut out from one face of a cubical wooden block such that the diameter l of the hemisphere is equal to the edge of the cube. Determine the surface area of the remaining solid.
Sol. Let ‘l’ be the side of the cube.
∴ The greatest diameter of the curved hemisphere = l
⇒ Radius of the curved hemisphere  
∴ Surface area of hemisphere = 2πr²

Base area of the hemisphere  
Surface area of the cube = 6 × l² = 6l²
∴ Surface area of the remaining solid



Q6. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends (see Fig.). The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Sol. Radius of the hemispherical part
= 5/2 mm = 2.5 mm
∴ Surface area of one hemispherical part = 2πr²
⇒ Surface area of both hemispherical parts

Area of cylindrical part =
∴ Total surface area

Q7. 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 m². (Note that the base of the tent will not be covered with canvas.)
Sol. 

For cylindrical part:
 Radius (r) = 4/2 m = 2 m
 Height (h) = 2.1 m
∴ Curved surface area = 2πrh =  
For conical part:
 Slant height (l) = 2.8 m
 Base radius (r)= 2m
∴ Curved surface area  
∴ Total surface area 
= [Surface area of the cylindrical part] + [Surface area of conical part]

Cost of the canvas used 
Cost of 1 m² of canvas = Rs 500
∴ Cost of 44 m² of canvas = Rs 500 × 44 = Rs. 22000

Q8. 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 cm².
Sol. 

For cylindrical part:
 Height = 2.4 cm
 Diameter = 1.4 cm
⇒ Radius (r) = 0.7 cm
For conical part:
 Base area (r) = 0.7 cm
 Height (h) = 2.4 cm
∴ Slant height
∴ Curved surface area of the conical part

Base area of the conical part

Total surface area of the remaining solid:

Q9. A wooden article was made by scooping out a hemisphere from each end of a solid cylinder, as shown in Fig. 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.

Sol. Radius of the cylinder (r) = 3.5 cm
Height of the cylinder (h) = 10 cm
∴ Total surface area = 2πrh + 2πr² = 2πr (h + r)

Curved surface area of a hemisphere = 2πr²
∴ Curved surface area of both hemispheres

Base area of a hemisphere = πr²

∴ Base area of both hemispheres = 2πr²

∴ Total surface area of the remaining solid
= 297 cm² + 154 cm² − 77 cm² 
= (451 − 77) cm² = 374 cm².