Volume of Cuboid
A cuboid is a three-dimensional solid bounded by six rectangular faces. It has 8 vertices, 12 edges, and 6 faces. Every face is a rectangle, and opposite faces are congruent.
The volume of a cuboid tells us the amount of space enclosed within it. It is measured in cubic units such as cm³, m³, or litres.
Common examples of cuboids include bricks, books, boxes, rooms, swimming pools, and water tanks. Calculating volume is essential for determining capacity, storage, and material requirements.
The volume formula is one of the most frequently used formulas in Class 8 mensuration and forms the basis for understanding volumes of more complex solids.
What is Volume of Cuboid?
Definition: A cuboid is a solid with six rectangular faces, where opposite faces are equal and parallel.
Dimensions of a cuboid:
- Length (l) — the longest edge of the base
- Breadth (b) — the shorter edge of the base
- Height (h) — the vertical edge
The three dimensions are mutually perpendicular (at right angles to each other).
Important: A cube is a special cuboid where l = b = h.
Volume of Cuboid Formula
Volume of a Cuboid:
Volume = l × b × h
Where:
- l = length
- b = breadth
- h = height
Related formulas:
- Total Surface Area = 2(lb + bh + lh)
- Lateral Surface Area = 2h(l + b)
- Diagonal of cuboid = √(l² + b² + h²)
Unit conversion:
- 1 m³ = 1,000 litres = 10,00,000 cm³
- 1 litre = 1,000 cm³
- 1 cm³ = 1 mL
Derivation and Proof
Derivation of Volume = l × b × h:
Step 1: The base of a cuboid is a rectangle with length l and breadth b.
Step 2: Area of the base = l × b.
Step 3: The cuboid is formed by stacking layers of this base rectangle up to height h.
Step 4: Volume = Area of base × height = l × b × h.
Using unit cubes:
- A cuboid of dimensions l × b × h (in cm) can hold l × b unit cubes in each layer.
- There are h such layers stacked vertically.
- Total unit cubes = l × b × h.
- Since each unit cube has volume 1 cm³, Volume = l × b × h cm³.
Types and Properties
Problems on volume of a cuboid can be classified as follows:
1. Finding volume from dimensions:
- Direct application of V = l × b × h.
2. Finding a missing dimension:
- Given volume and two dimensions, find the third: h = V / (l × b).
- Finding how many litres a tank can hold (convert cm³ to litres).
4. Packing problems:
- How many smaller cuboids fit inside a larger one.
5. Cost problems:
- Finding cost of filling, painting, or constructing a cuboid.
6. Comparison problems:
- Comparing volumes of two cuboids or a cuboid and a cube.
Solved Examples
Example 1: Example 1: Direct computation
Problem: Find the volume of a cuboid with length 12 cm, breadth 8 cm, and height 5 cm.
Solution:
Given:
- l = 12 cm, b = 8 cm, h = 5 cm
Using the formula:
- Volume = l × b × h
- Volume = 12 × 8 × 5
- Volume = 480 cm³
Answer: The volume is 480 cm³.
Example 2: Example 2: Finding a missing dimension
Problem: A cuboid has volume 1,200 cm³. Its length is 15 cm and breadth is 10 cm. Find the height.
Solution:
- V = l × b × h
- 1,200 = 15 × 10 × h
- 1,200 = 150 × h
- h = 1,200 / 150 = 8 cm
Answer: The height is 8 cm.
Example 3: Example 3: Water tank capacity
Problem: A water tank is 2 m long, 1.5 m wide, and 1 m deep. How many litres of water can it hold?
Solution:
- Volume = 2 × 1.5 × 1 = 3 m³
- 1 m³ = 1,000 litres
- Capacity = 3 × 1,000 = 3,000 litres
Answer: The tank can hold 3,000 litres.
Example 4: Example 4: Packing boxes
Problem: A carton is 60 cm long, 40 cm wide, and 30 cm high. How many boxes of dimensions 10 cm × 10 cm × 10 cm can be packed in it?
Solution:
- Volume of carton = 60 × 40 × 30 = 72,000 cm³
- Volume of one box = 10 × 10 × 10 = 1,000 cm³
- Number of boxes = 72,000 / 1,000 = 72
Answer: 72 boxes can be packed.
Example 5: Example 5: Cost of filling
Problem: A swimming pool is 25 m long, 10 m wide, and 2 m deep. If the cost of filling water is ₹0.50 per litre, find the total cost.
Solution:
- Volume = 25 × 10 × 2 = 500 m³
- In litres = 500 × 1,000 = 5,00,000 litres
- Cost = 5,00,000 × 0.50 = ₹2,50,000
Answer: The total cost is ₹2,50,000.
Example 6: Example 6: Finding volume in litres
Problem: A box is 50 cm long, 30 cm wide, and 20 cm high. Find its volume in litres.
Solution:
- Volume = 50 × 30 × 20 = 30,000 cm³
- Volume in litres = 30,000 / 1,000 = 30 litres
Answer: The volume is 30 litres.
Example 7: Example 7: Comparing volumes
Problem: Cuboid A has dimensions 8 cm × 6 cm × 4 cm. Cuboid B has dimensions 12 cm × 4 cm × 3 cm. Which has a greater volume?
Solution:
- Volume of A = 8 × 6 × 4 = 192 cm³
- Volume of B = 12 × 4 × 3 = 144 cm³
Answer: Cuboid A has the greater volume (192 cm³ > 144 cm³).
Example 8: Example 8: Melting and recasting
Problem: A metal block of dimensions 20 cm × 10 cm × 5 cm is melted and recast into small cubes of side 2 cm. How many cubes are formed?
Solution:
- Volume of block = 20 × 10 × 5 = 1,000 cm³
- Volume of one cube = 2 × 2 × 2 = 8 cm³
- Number of cubes = 1,000 / 8 = 125
Answer: 125 cubes are formed.
Example 9: Example 9: Room volume
Problem: A room is 6 m long, 4 m wide, and 3 m high. Find the volume of air in the room in cubic metres.
Solution:
- Volume = 6 × 4 × 3 = 72 m³
Answer: The room contains 72 m³ of air.
Example 10: Example 10: Increasing one dimension
Problem: A cuboid has dimensions 10 cm × 8 cm × 6 cm. If the length is doubled, by how much does the volume increase?
Solution:
- Original volume = 10 × 8 × 6 = 480 cm³
- New volume = 20 × 8 × 6 = 960 cm³
- Increase = 960 − 480 = 480 cm³
The volume also doubles when the length is doubled.
Answer: The volume increases by 480 cm³ (it doubles).
Real-World Applications
Construction: Calculating the volume of rooms, walls, and foundations to estimate material requirements such as concrete, plaster, and paint.
Packaging: Determining how many items can fit in a carton or shipping container. Logistics companies optimise packing using volume calculations.
Water Storage: Tanks, reservoirs, and swimming pools are cuboid-shaped. Volume calculations determine the water capacity in litres.
Agriculture: Granaries and storage bins are often cuboidal. Volume helps estimate how much grain or produce can be stored.
Daily Life: Calculating the volume of a refrigerator, an aquarium, a suitcase, or a shoebox to determine storage capacity.
Key Points to Remember
- A cuboid has 6 rectangular faces, 8 vertices, and 12 edges.
- Volume of cuboid = l × b × h.
- Volume is measured in cubic units (cm³, m³).
- 1 m³ = 1,000 litres; 1 litre = 1,000 cm³.
- A cube is a special cuboid where l = b = h; its volume = side³.
- Total Surface Area = 2(lb + bh + lh).
- Lateral Surface Area = 2h(l + b).
- If one dimension is doubled, the volume doubles.
- If all dimensions are doubled, the volume becomes 8 times the original.
- Volume of cuboid = Base area × height.
Practice Problems
- Find the volume of a cuboid with l = 14 cm, b = 10 cm, h = 6 cm.
- A cuboid has volume 2,160 cm³. Its length is 18 cm and height is 12 cm. Find the breadth.
- A tank is 3 m × 2 m × 1.5 m. How many litres of water can it hold?
- How many cubes of side 3 cm can be cut from a cuboid of dimensions 18 cm × 12 cm × 9 cm?
- A brick is 20 cm × 10 cm × 8 cm. Find the volume of 500 bricks in m³.
- If all dimensions of a cuboid are tripled, by what factor does the volume increase?
- A room is 5 m long, 4 m wide, and 3 m high. Find the cost of filling it with gas at ₹2 per m³.
- A cuboid and a cube have the same volume. The cuboid is 16 cm × 8 cm × 4 cm. Find the side of the cube.
Frequently Asked Questions
Q1. What is the formula for the volume of a cuboid?
Volume = length × breadth × height, or V = l × b × h. The result is in cubic units.
Q2. What is the difference between a cube and a cuboid?
A cube has all edges equal (l = b = h) and all faces are squares. A cuboid has three different pairs of edges and all faces are rectangles. A cube is a special case of a cuboid.
Q3. How do you convert cm³ to litres?
Divide by 1,000. For example, 5,000 cm³ = 5,000 / 1,000 = 5 litres.
Q4. How do you convert m³ to litres?
Multiply by 1,000. For example, 2.5 m³ = 2.5 × 1,000 = 2,500 litres.
Q5. What happens to the volume if all dimensions are doubled?
The volume becomes 2 × 2 × 2 = 8 times the original. In general, if all dimensions are multiplied by k, the volume becomes k³ times.
Q6. What is the diagonal of a cuboid?
The space diagonal (from one corner to the diagonally opposite corner) = √(l² + b² + h²).
Q7. Can the volume of a cuboid be negative?
No. Length, breadth, and height are all positive quantities, so the volume is always positive.
Q8. What is the difference between volume and surface area?
Volume measures the space inside the cuboid (in cubic units). Surface area measures the total area of all six faces (in square units). They are different quantities with different units.
Q9. How do you find volume if only base area and height are given?
Volume = Base area × height. The base area equals l × b, so this is equivalent to l × b × h.
Q10. What is 1 mL in cm³?
1 mL = 1 cm³. This is an exact equivalence.
