Cube and Cuboid: Volume, Surface Area, Diagonal, Rooms and Bricks
Cube and Cuboid: Volume, Surface Area, Diagonal, Rooms and Bricks
Cube and cuboid questions need four formulas: volume, total surface area, lateral (four walls) area and diagonal. NTPC adds edge ratios with a given sum of edges, room-painting costs and how many small boxes or bricks fit in a bigger space.
1. Formula Box
| Solid | Volume | Total surface area | Lateral area (4 walls) | Diagonal |
|---|---|---|---|---|
| Cuboid (l, b, h) | lbh | 2(lb + bh + hl) | 2(l + b)h | √(l² + b² + h²) |
| Cube (edge a) | a³ | 6a² | 4a² | a√3 |
Sum of all 12 edges of a cuboid = 4(l + b + h)
(l + b + h)² = (l² + b² + h²) + 2(lb + bh + hl) = diagonal² + TSA
Number of small boxes in a big box = big volume ÷ small volume (when the dimensions divide evenly)
1 m³ = 1,000 litres = 10⁶ cm³
✗ A cube's edge rises 20%, so its volume rises 60% | ✓ Volume factor = 1.2³ = 1.728, a 72.8% rise
✗ Painting a room's four walls = TSA | ✓ Four walls = 2(l + b)h; the ceiling and floor are separate
हिंदी नोट: घनाभ का आयतन lbh, कुल पृष्ठीय क्षेत्रफल 2(lb + bh + hl) और चार दीवारों का क्षेत्रफल 2(l + b)h होता है। घन के लिए a³, 6a² और विकर्ण a√3।
Exam Pointer: Verified NTPC patterns: cube volume from its diagonal (January 2021), cuboid edges in a ratio with the edge sum or volume given (April 2016, June 2022 CBT-2) and how many small cuboids fit in a room (January 2021).
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Cube: diagonal, volume and surface area
[PYQ: NTPC CBT-1 19-Jan-2021 Shift-2]
EXAM LEVEL
Q. The diagonal of a cube is 6√3 cm. Find its volume and total surface area.
Edge = 6 cm. Volume = 216 cm³; TSA = 6 × 36 = 216 cm².
Answer: 216 cm³ and 216 cm²
EXAMATLAS LEVEL
Q. The total surface area of a cube is 1,350 cm². Find its volume, and the volume if each edge is increased by 20%.
6a² = 1,350 gives a² = 225 and a = 15 cm, volume 3,375 cm³. New edge 18 cm: volume 5,832 cm³ (= 3,375 × 1.728).
Answer: 3,375 cm³; 5,832 cm³
Pattern 2: Cuboid: edges in a ratio, surface area and volume
[PYQ: NTPC CBT-2 12-Jun-2022 Shift-1 | NTPC CBT-1 9-Apr-2016 Shift-3]
EXAM LEVEL
Q. Find the volume, total surface area and diagonal of a cuboid 12 cm × 8 cm × 5 cm.
Volume = 480 cm³. TSA = 2(96 + 40 + 60) = 392 cm². Diagonal = √(144 + 64 + 25) = √233 ≈ 15.26 cm.
Answer: 480 cm³; 392 cm²; √233 cm
EXAMATLAS LEVEL
Q. The length, breadth and height of a cuboid are in the ratio 3 : 2 : 1, and the sum of all its edges is 96 cm. Find its total surface area and volume.
4(l + b + h) = 96, so l + b + h = 24 and the edges are 12, 8, 4 cm. TSA = 2(96 + 32 + 48) = 352 cm². Volume = 384 cm³.
Answer: 352 cm²; 384 cm³
Pattern 3: Rooms, walls, bricks and boxes
[PYQ: NTPC CBT-1 5-Jan-2021 Shift-1]
EXAM LEVEL
Q. A room is 8 m long, 6 m wide and 4 m high. Find the cost of painting its four walls at ₹15 per m².
Four walls = 2(8 + 6) × 4 = 112 m². Cost = 112 × 15 = ₹1,680.
Answer: ₹1,680
EXAMATLAS LEVEL
Q. How many bricks of 25 cm × 20 cm × 10 cm are needed for a wall 10 m long, 30 cm thick and 2 m high, if mortar takes up 10% of the wall's volume?
Wall = 1,000 × 30 × 200 = 60,00,000 cm³; bricks fill 90% = 54,00,000 cm³. One brick = 5,000 cm³. Number = 54,00,000/5,000 = 1,080.
Answer: 1,080 bricks
Pattern 4: Cutting and melting cubes
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. A cube of edge 12 cm is cut into cubes of edge 3 cm. How many small cubes are formed, and how many times the original surface area is their total surface area?
Number = (12/3)³ = 64. Original TSA = 864 cm²; each small cube has 54 cm², total 3,456 cm², which is 4 times (the edge ratio 12/3).
Answer: 64 cubes; 4 times
EXAMATLAS LEVEL
Q. Three metal cubes of edges 3 cm, 4 cm and 5 cm are melted into one cube. Find its edge and the ratio of its surface area to the total surface area of the three cubes.
Volume = 27 + 64 + 125 = 216 cm³, so the edge is 6 cm. New TSA = 216 cm²; old total = 54 + 96 + 150 = 300 cm². Ratio = 216 : 300 = 18 : 25.
Answer: 6 cm; 18 : 25
60-Second Revision
- Cuboid: lbh, 2(lb + bh + hl), walls 2(l + b)h, diagonal √(l² + b² + h²).
- Cube: a³, 6a², diagonal a√3.
- Edge sum = 4(l + b + h); edge ratio k gives volume ratio k³.
- Count boxes or bricks by volume; remove mortar first.