Volume & Surface Areas
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Solved Examples
Study MaterialSolved Examples – Volume & Surface Areas
Solved examples help students understand the practical application of mensuration formulas and 3D geometry concepts in competitive examinations. These examples are designed from basic to advanced level and cover important questions frequently asked in SSC, Banking, Railway, CDS, NDA, CAT, Defence, and various aptitude examinations.
Topics Covered in Solved Examples
- Volume of Cube and Cuboid
- Surface Area Problems
- Cylinder and Cone Questions
- Sphere and Hemisphere Problems
- Water Tank Problems
- Melting and Recasting Solids
- CSA and TSA Concepts
- Advanced Mensuration Problems
Example 1: Volume of Cube
Question: Find the volume of a cube whose side is 8 cm.
Solution:
Volume of cube:
= a3
= 83
= 512 cm3
Answer: 512 cm3
Example 2: Total Surface Area of Cube
Question: Find the total surface area of a cube whose side is 6 cm.
Solution:
Total Surface Area:
= 6a2
= 6 × 62
= 6 × 36
= 216 cm2
Answer: 216 cm2
Example 3: Volume of Cuboid
Question: Find the volume of a cuboid whose length, breadth, and height are 12 cm, 8 cm, and 5 cm respectively.
Solution:
Volume:
= l × b × h
= 12 × 8 × 5
= 480 cm3
Answer: 480 cm3
Example 4: Diagonal of Cuboid
Question: Find the diagonal of a cuboid whose dimensions are 3 cm, 4 cm, and 12 cm.
Solution:
Diagonal:
= √(l2 + b2 + h2)
= √(32 + 42 + 122)
= √(9 + 16 + 144)
= √169
= 13 cm
Answer: 13 cm
Example 5: Volume of Cylinder
Question: Find the volume of a cylinder whose radius is 7 cm and height is 10 cm.
Solution:
Volume:
= πr2h
= 22/7 × 7 × 7 × 10
= 1540 cm3
Answer: 1540 cm3
Example 6: Curved Surface Area of Cylinder
Question: Find the curved surface area of a cylinder whose radius is 14 cm and height is 8 cm.
Solution:
CSA:
= 2Ï€rh
= 2 × 22/7 × 14 × 8
= 704 cm2
Answer: 704 cm2
Example 7: Volume of Cone
Question: Find the volume of a cone whose radius is 6 cm and height is 7 cm.
Solution:
Volume:
= (1/3)Ï€r2h
= (1/3) × 22/7 × 6 × 6 × 7
= 264 cm3
Answer: 264 cm3
Example 8: Slant Height of Cone
Question: Find the slant height of a cone whose radius is 5 cm and height is 12 cm.
Solution:
Slant Height:
= √(r2 + h2)
= √(52 + 122)
= √(25 + 144)
= √169
= 13 cm
Answer: 13 cm
Example 9: Volume of Sphere
Question: Find the volume of a sphere whose radius is 7 cm.
Solution:
Volume:
= (4/3)Ï€r3
= (4/3) × 22/7 × 7 × 7 × 7
= 1437.33 cm3
Answer: 1437.33 cm3
Example 10: Surface Area of Sphere
Question: Find the surface area of a sphere whose radius is 14 cm.
Solution:
Surface Area:
= 4Ï€r2
= 4 × 22/7 × 14 × 14
= 2464 cm2
Answer: 2464 cm2
Example 11: Volume of Hemisphere
Question: Find the volume of a hemisphere whose radius is 6 cm.
Solution:
Volume:
= (2/3)Ï€r3
= (2/3) × 22/7 × 6 × 6 × 6
= 452.57 cm3
Answer: 452.57 cm3
Example 12: Total Surface Area of Hemisphere
Question: Find the total surface area of a hemisphere whose radius is 7 cm.
Solution:
TSA:
= 3Ï€r2
= 3 × 22/7 × 7 × 7
= 462 cm2
Answer: 462 cm2
Example 13: Water Tank Problem
Question: A water tank is 5 m long, 4 m broad, and 3 m high. Find its capacity.
Solution:
Volume:
= l × b × h
= 5 × 4 × 3
= 60 m3
Answer: 60 m3
Example 14: Melting and Recasting Problem
Question: A cube of side 6 cm is melted and recast into smaller cubes of side 2 cm. Find the number of smaller cubes formed.
Solution:
Volume of big cube:
= 63
= 216 cm3
Volume of small cube:
= 23
= 8 cm3
Number of cubes:
= 216 / 8
= 27
Answer: 27 cubes
Example 15: Hollow Cylinder Problem
Question: Find the volume of a hollow cylinder whose external radius is 7 cm, internal radius is 5 cm, and height is 10 cm.
Solution:
Volume:
= πh(R2 − r2)
= 22/7 × 10 × (49 − 25)
= 22/7 × 10 × 24
= 754.29 cm3
Answer: 754.29 cm3
Example 16: CSA vs TSA Problem
Question: Find the total surface area of a cylinder whose radius is 7 cm and height is 20 cm.
Solution:
TSA:
= 2Ï€r(h+r)
= 2 × 22/7 × 7 × (20 + 7)
= 44 × 27
= 1188 cm2
Answer: 1188 cm2
Example 17: Cube Surface Area Ratio
Question: If the side ratio of two cubes is 2:3, find the ratio of their surface areas.
Solution:
Surface area ratio:
= 22 : 32
= 4 : 9
Answer: 4 : 9
Example 18: Cube Volume Ratio
Question: If the side ratio of two cubes is 3:5, find the ratio of their volumes.
Solution:
Volume ratio:
= 33 : 53
= 27 : 125
Answer: 27 : 125
Important Exam Tips
- Memorize all mensuration formulas carefully.
- Practice CSA and TSA concepts regularly.
- Use π = 22/7 whenever suitable.
- Learn important Pythagorean triplets.
- Draw diagrams for better visualization.
- Verify units carefully.
- Practice melting and recasting problems thoroughly.
Practicing solved examples regularly improves conceptual clarity, logical thinking, and calculation speed in solving mensuration aptitude questions in competitive examinations.