Class 10 Mathematics
Chapter 12: Surface Areas and Volumes
Exercise 12.1 – Stepwise Solutions
Instruction: Unless stated otherwise, take π = 22/7.
Q1) Two cubes each of volume 64 cm³ are joined end to end. Find the surface area of the resulting cuboid.
Step 1: Volume of cube = a³
a³ = 64
a = 4 cm
Step 2: When two cubes are joined end-to-end:
Length = 4 + 4 = 8 cm
Breadth = 4 cm
Height = 4 cm
Step 3: Surface area of cuboid = 2(lb + bh + lh)
= 2(8×4 + 4×4 + 8×4)
= 2(32 + 16 + 32)
= 2 × 80
Surface area = 160 cm²
Q2) A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder.
Diameter of hemisphere = 14 cm
Radius r = 7 cm
Total height of vessel = 13 cm
Step 1: Height of cylinder
Height = 13 − 7 = 6 cm
Step 2: Inner surface area = CSA of cylinder + CSA of hemisphere
CSA cylinder = 2πrh
= 2 × 22/7 × 7 × 6
= 264 cm²
CSA hemisphere = 2πr²
= 2 × 22/7 × 7 × 7
= 308 cm²
Total inner surface area
= 264 + 308
= 572 cm²
Q3) A toy is in the form of a cone mounted on a hemisphere.
Radius r = 3.5 cm
Total height = 15.5 cm
Step 1: Height of cone
Height of hemisphere = r = 3.5
Height of cone = 15.5 − 3.5
= 12 cm
Step 2: Slant height of cone
l = √(h² + r²)
= √(12² + 3.5²)
= √(144 + 12.25)
= √156.25
= 12.5 cm
Step 3: Total surface area
= CSA cone + CSA hemisphere
CSA cone = πrl
= 22/7 × 3.5 × 12.5
= 137.5 cm²
CSA hemisphere = 2πr²
= 2 × 22/7 × 3.5²
= 77 cm²
Total surface area
= 137.5 + 77
= 214.5 cm²
Q4) A cubical block of side 7 cm is surmounted by a hemisphere.
Step 1: Greatest diameter of hemisphere = side of cube
= 7 cm
Radius r = 3.5 cm
Step 2: Surface area
Surface area of cube = 6a²
= 6 × 7²
= 6 × 49
= 294 cm²
Area of circular base covered by hemisphere
= πr²
= 22/7 × 3.5²
= 38.5 cm²
Remaining cube area
= 294 − 38.5
= 255.5 cm²
CSA hemisphere = 2πr²
= 2 × 38.5
= 77 cm²
Total surface area
= 255.5 + 77
= 332.5 cm²
Q5) Hemispherical depression cut from cube.
Edge of cube = diameter of hemisphere = l
Radius r = l/2
Step 1: Surface area of cube
= 6l²
Step 2: Area removed = πr²
Step 3: Curved surface area of hemisphere added
= 2πr²
Total surface area
= 6l² − πr² + 2πr²
= 6l² + πr²
Q6) A medicine capsule consists of cylinder and two hemispheres.
Total length = 14 mm
Diameter = 5 mm
Radius r = 2.5 mm
Step 1: Length of cylindrical part
= 14 − 5
= 9 mm
Step 2: Surface area
= CSA cylinder + CSA sphere
CSA cylinder = 2πrh
= 2 × 22/7 × 2.5 × 9
= 141.43 mm²
Surface area sphere = 4πr²
= 4 × 22/7 × 2.5²
= 78.57 mm²
Total surface area
= 141.43 + 78.57
= 220 mm² (approx)
Q7) Tent shaped like cylinder with conical top.
Radius r = 2 m
Height of cylinder = 2.1 m
Slant height cone = 2.8 m
Step 1: Canvas area
= CSA cylinder + CSA cone
CSA cylinder = 2πrh
= 2 × 22/7 × 2 × 2.1
= 26.4 m²
CSA cone = πrl
= 22/7 × 2 × 2.8
= 17.6 m²
Total canvas area
= 26.4 + 17.6
= 44 m²
Cost
= 44 × 500
= ₹22000
Q8) Cylinder with conical cavity.
Height = 2.4 cm
Diameter = 1.4 cm
Radius r = 0.7 cm
Step 1: Slant height of cone
l = √(h² + r²)
= √(2.4² + 0.7²)
= √(5.76 + 0.49)
= √6.25
= 2.5 cm
Step 2: Total surface area
= CSA cylinder + base area + CSA cone
CSA cylinder = 2πrh
= 2 × 22/7 × 0.7 × 2.4
= 10.56 cm²
Base area = πr²
= 22/7 × 0.49
= 1.54 cm²
CSA cone = πrl
= 22/7 × 0.7 × 2.5
= 5.5 cm²
Total surface area
= 10.56 + 1.54 + 5.5
= 17.6 cm² ≈ 18 cm²
Q9) Wooden article formed by scooping hemisphere from both ends.
Radius r = 3.5 cm
Height of cylinder = 10 cm
Step 1: Curved surface area cylinder
= 2πrh
= 2 × 22/7 × 3.5 × 10
= 220 cm²
Step 2: Area of two hemispherical hollows
= 2 × (2πr²)
= 4πr²
= 4 × 22/7 × 3.5²
= 154 cm²
Total surface area
= 220 + 154
= 374 cm²
Final Answers Summary
- Q1 → 160 cm²
- Q2 → 572 cm²
- Q3 → 214.5 cm²
- Q4 → 332.5 cm²
- Q6 → 220 mm²
- Q7 → Area 44 m², Cost ₹22000
- Q8 → 18 cm²
- Q9 → 374 cm²