Volume of Cube and Cuboids
Pointwise AISSEE Class 6 Mathematics guide for Volume of Cube and Cuboids (Topic 18). Master volume formulas (a³ and l×b×h), Total Surface Area, Area of 4 walls, liters-to-cm³ capacity, and space diagonals.
Overview & Fundamental Concept
Pointwise AISSEE Class 6 Mathematics guide for Volume of Cube and Cuboids (Topic 18). Master volume formulas (a³ and l×b×h), Total Surface Area, Area of 4 walls, liters-to-cm³ capacity, and space diagonals.
Official Syllabus Coverage Breakdown
| Syllabus Item | Prescribed Scope | Exam Relevance & Pedagogical Objective |
|---|---|---|
| Item 18 | Concept of 3D volume, cubic units, volume of a cube, volume of a cuboid, surface area of cube and cuboid, capacity in liters. | Master 3D spatial mensuration, cube and cuboid volume/surface area formulas, liquid capacity conversions (liters to cm³), and room diagonal calculations for AISSEE Class VI. |
1. 3D Spatial Geometry: Faces, Edges & Vertices
- Concept of Volume: Volume is the measure of the 3-dimensional space occupied by a solid body. Measured in cubic units ($\text{cm}^3, \text{m}^3$).
- Capacity: The interior volume of a hollow container, representing the maximum amount of liquid it can hold. Expressed in liters ($L$) or milliliters ($mL$).
- Universal Liquid-Volume Conversion Standards:
• $1\text{ Liter} = 1,000\text{ cm}^3$
• $1\text{ m}^3 = 1,000\text{ Liters} = 1,000,000\text{ cm}^3$
• $1\text{ mL} = 1\text{ cm}^3$ - Euler's Polyhedron Properties: Both cube and cuboid have: 6 Faces, 12 Edges, and 8 Vertices ($F + V - E = 6 + 8 - 12 = 2$).
| Attribute | Cube (Side $a$) | Cuboid (Length $l$, Breadth $b$, Height $h$) |
|---|---|---|
| Faces | 6 identical square faces | 6 rectangular faces (opposite faces identical) |
| Edges | 12 equal edges ($a$) | 12 edges (4 lengths, 4 breadths, 4 heights) |
| Vertices | 8 corner vertices | 8 corner vertices |
| Volume | $V = a^3$ | $V = l \times b \times h$ |
| Total Surface Area (TSA) | $6a^2$ | $2(lb + bh + hl)$ |
| Lateral Surface Area (LSA) | $4a^2$ | $2(l + b)h$ (Area of 4 walls) |
| Space Diagonal | $d = a\sqrt{3}$ | $d = \sqrt{l^2 + b^2 + h^2}$ |
Cadet Mnemonic • 6-8-12 Solid Rule
Remember solid vertices and edges:
- 6 Faces
- 8 Vertices
- 12 Edges
2. Cube & Cuboid Mensuration Formulas
- Volume Formulas:
• $\text{Volume of Cuboid} = \text{Length} \times \text{Breadth} \times \text{Height} = l \times b \times h$
• $\text{Volume of Cube} = (\text{Edge})^3 = a^3$ - Surface Area Formulas:
• $\text{TSA of Cuboid} = 2(lb + bh + hl)$
• $\text{Area of Four Walls (LSA)} = 2(l + b) \times h = \text{Perimeter of Base} \times \text{Height}$
• $\text{TSA of Cube} = 6a^2$
• $\text{LSA of Cube} = 4a^2$ - Longest Rod in a Room: The length of the longest pole/rod that can be placed inside a rectangular room is equal to its space diagonal: $$\text{Longest Rod} = \sqrt{l^2 + b^2 + h^2}$$ For a cube: $\text{Longest Rod} = a\sqrt{3}$.
Exam Trick • Longest Rod in a Room
For a room measuring 4 m × 3 m × 12 m: Longest rod = √(4² + 3² + 12²) = √(16 + 9 + 144) = √169 = 13 meters!
AISSEE Trap Alert • Area of 4 Walls vs TSA
Area of 4 walls does NOT include the roof and the floor! Formula is 2(l + b)h. If whitewashing walls AND ceiling: Area = 2(l + b)h + lb.
3. Melting, Recasting & Brick Count Algorithms
- Conservation of Volume in Melting: When a solid metal block is melted and recast into smaller solids, the total volume remains constant: $$\text{Total Volume Before Melting} = \text{Total Volume After Recasting}$$
- Brick / Box Count Formula: To find how many smaller bricks can fit inside a wall or container: $$\text{Number of Bricks} = \frac{\text{Volume of Wall}}{\text{Volume of One Brick}}$$ Crucial: Convert all dimensions of wall and brick into the exact SAME unit (centimeters) before computing!
Visual Concept Blueprint • Vector Architecture
Speed Shortcuts & Mental Calculation Strategies
Master these high-yield time-saving heuristics curated by top AISSEE scorers to eliminate calculation bottlenecks and answer questions in under 45 seconds.
Shortcut Technique 1: Side from Cube Volume
If volume of a cube is $V$, its side is the cube root: $a = \sqrt[3]{V}$. For $V = 216\text{ cm}^3$, side $a = 6\text{ cm}$.
Shortcut Technique 2: Liquid Tank Capacity in Liters
To find capacity of a water tank in liters: Compute volume in $\text{m}^3$ and multiply by $1,000$, or volume in $\text{cm}^3$ and divide by $1,000$.
Shortcut Technique 3: Edge Doubling Effect on Volume
If the edge of a cube is doubled ($2a$), its volume becomes $2^3 = 8$ times greater! Surface area becomes $2^2 = 4$ times greater.
Interactive 3D Revision Flashcards Deck
Click any card to flip and reveal the key exam rule or topper secret.
REVISION FLASHCARD
What is the core rule & definition for Volume of Cube and Cuboids?
CADET MEMORY KEY
Volume of Cuboid: V = l \times b \times h
REVISION FLASHCARD
Cadet Shortcut: How to solve Volume of Cube and Cuboids in under 30 seconds?
CADET MEMORY KEY
Fast MCQ Elimination: Check the last digit (unit digit) and estimate the order of magnitude to eliminate 2 distractors in under 10 seconds.
REVISION FLASHCARD
Exam Trap Alert: What pitfall do most candidates fall into?
CADET MEMORY KEY
Cubic Units: Volume is always measured in cubic units (cm³, m³).
REVISION FLASHCARD
Rapid Memory Anchor for AISSEE Hall Recall:
CADET MEMORY KEY
Surface Area: Total Surface Area of Cube = 6a². Cuboid = 2(lb + bh + hl).
REVISION FLASHCARD
What is the volume of a cube whose total surface area is 96 cm²?
CADET MEMORY KEY
TSA = 6a² = 96 => a² = 16 => a = 4 cm. Volume = a³ = 4³ = 64 cm³.
Common Traps & Examination Pitfalls
Avoid the Speed-Guess Trap: Under AISSEE pressure, candidates frequently misread units (e.g., mixing metres and centimetres, or per hour and per second) and confuse complementary ($90^\circ$) with supplementary ($180^\circ$) angles. Read every stem carefully and check units before selecting your OMR option.
Key Identities & Formulas
Cuboid Volume
$$V = l \times b \times h$$
Product of length, breadth, and height.
Cube Volume
$$V = a^3$$
Cube of edge length.
Longest Diagonal
$$d = \sqrt{l^2 + b^2 + h^2}$$
Longest pole that fits in a rectangular room.
Step-by-Step Solved Examination Models
Solved Example 1
Model Problem
Find the volume of a wooden box of dimensions $12\text{ cm} \times 8\text{ cm} \times 5\text{ cm}$.
Detailed Solution:
Volume of a cuboid = $\text{length} \times \text{breadth} \times \text{height} = 12 \times 8 \times 5 = 480\text{ cm}^3$.
Solved Example 2
Model Problem
How many bricks of size $20\text{ cm} \times 10\text{ cm} \times 5\text{ cm}$ are required to build a wall of size $4\text{ m} \times 3\text{ m} \times 20\text{ cm}$?
Detailed Solution:
Convert all to cm: Wall = $400 \times 300 \times 20 = 2,400,000\text{ cm}^3$.
Volume of 1 brick = $20 \times 10 \times 5 = 1,000\text{ cm}^3$.
Bricks needed = $2,400,000 / 1,000 = 2,400$ bricks.
Solved Example 3
Model Problem
Find the length of the longest measuring rod that can be placed inside a rectangular room measuring 8 m long, 6 m wide, and 10 m high.
Detailed Solution:
1. The longest rod that can fit in a room equals the space diagonal: $d = \sqrt{l^2 + b^2 + h^2}$.
2. $d = \sqrt{8^2 + 6^2 + 10^2} = \sqrt{64 + 36 + 100} = \sqrt{200}$.
3. Simplify $\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\text{ meters}$. Correct Option is (A).
2. $d = \sqrt{8^2 + 6^2 + 10^2} = \sqrt{64 + 36 + 100} = \sqrt{200}$.
3. Simplify $\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\text{ meters}$. Correct Option is (A).
AISSEE High-Yield Practice Drills (10 Questions)
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Question 1 of 10
Exam Standard
Find the volume of a cube whose edge measures 6 cm.
Correct Answer: Option (A)
Volume of cube = $\text{side}^3 = 6 \times 6 \times 6 = 216\text{ cm}^3$.
Question 2 of 10
Exam Standard
A water tank is $3\text{ m}$ long, $2\text{ m}$ wide, and $1.5\text{ m}$ deep. What is its capacity in litres?
Correct Answer: Option (A)
Volume = $3 \times 2 \times 1.5 = 9\text{ m}^3$. Since $1\text{ m}^3 = 1,000\text{ litres}$, capacity = $9 \times 1,000 = 9,000\text{ litres}$.
Question 3 of 10
Exam Standard
How many vertices, edges, and faces does a cube have?
Correct Answer: Option (A)
A standard cube has 6 square faces, 12 straight edges, and 8 corner vertices.
Question 4 of 10
Exam Standard
If the volume of a cube is $343\text{ cm}^3$, what is the length of its edge?
Correct Answer: Option (A)
Edge = $\sqrt[3]{343} = 7$ cm (since $7 \times 7 \times 7 = 343$).
Question 5 of 10
Exam Standard
Find the total surface area of a cube of edge 5 cm.
Correct Answer: Option (A)
Total surface area = $6 \times \text{edge}^2 = 6 \times (5 \times 5) = 6 \times 25 = 150\text{ sq cm}$.
Question 6 of 10
Exam Standard
A cuboidal block of iron has volume $720\text{ cm}^3$. If its length is 12 cm and breadth is 10 cm, find its height.
Correct Answer: Option (A)
Height = $\text{Volume} / (l \times b) = 720 / (12 \times 10) = 720 / 120 = 6$ cm.
Question 7 of 10
Exam Standard
How many small cubes of side 2 cm can be cut from a larger wooden cube of side 8 cm?
Correct Answer: Option (A)
Volume of large cube = $8^3 = 512\text{ cm}^3$. Volume of small cube = $2^3 = 8\text{ cm}^3$. Number of cubes = $512 / 8 = 64$.
Question 8 of 10
Exam Standard
What is the volume of a cuboid whose base area is 45 sq cm and height is 8 cm?
Correct Answer: Option (A)
Volume = $\text{Base Area} \times \text{height} = 45 \times 8 = 360\text{ cm}^3$.
Question 9 of 10
Exam Standard
If each edge of a cube is doubled, its volume becomes:
Correct Answer: Option (A)
New volume = $(2a)^3 = 8a^3$. It becomes 8 times the original volume.
Question 10 of 10
Exam Standard
A matchbox measures $4\text{ cm} \times 2.5\text{ cm} \times 1.5\text{ cm}$. What will be the volume of a packet containing 12 such matchboxes?
Correct Answer: Option (A)
Volume of 1 box = $4 \times 2.5 \times 1.5 = 15\text{ cm}^3$. Volume of 12 boxes = $15 \times 12 = 180\text{ cm}^3$.
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Last-Minute Cadet Revision Matrix
Rapid recall checkpoints to review 24 hours before stepping into the examination hall.
| Concept Checkpoint | High-Yield Exam Rule & Memory Anchor |
|---|---|
| Liters to cm³ | 1 Liter = 1,000 cm³; 1 m³ = 1,000 Liters. |
| Area of 4 Walls | Area of 4 walls = 2(l + b) × h. |
| TSA of Cube | Total Surface Area of cube = 6a²; Lateral Surface Area = 4a². |
| Solid Counts | Both cube and cuboid have 6 faces, 12 edges, and 8 vertices. |
Frequently Asked Questions
How many questions from Volume of Cube and Cuboids appear in the AISSEE Class VI exam?
Typically, 2 to 3 questions directly test Volume of Cube and Cuboids in the 50-question Mathematics paper, contributing 6 to 9 marks towards the 150-mark section total.
Is there any negative marking for incorrect answers in AISSEE Class VI?
No. The National Testing Agency (NTA) does not apply negative marking in the AISSEE Class VI entrance examination. Candidates should attempt every single question.
What is the best way to revise Volume of Cube and Cuboids before the exam?
Review the high-yield formulas, practice the 10 interactive drill MCQs on this page, and memorize the revision checkpoints matrix.