NCERT Solutions for Class 9 Maths Surface Area and Volume Ex 13.5

Understanding the Volume of a Cuboid

Imagine you have a rectangular sheet of paper. Its area is simply its length times its breadth. Now, what if you start stacking identical rectangular sheets one on top of the other? You create a 3D shape – a cuboid! The total space this stack occupies is its volume. This simple idea gives us the formula for the volume of a cuboid. The volume is the area of the base multiplied by the height.

Since the base of a cuboid is a rectangle with area = length × breadth, the formula for its volume becomes:
Volume of a Cuboid (V) = length × breadth × height
Or, V = l × b × h

A cube is just a special type of cuboid where the length, breadth, and height are all equal (let's call the side 'a').
So, for a cube, V = a × a × a = a³.

Volume is always measured in cubic units, such as cubic centimeters (cm³) or cubic meters (m³). This is because you are multiplying three length dimensions together.

Key Formulas and Conversions for Ex 13.5

Volume of a Cuboid

The space occupied by a cuboid. Formula: V = l × b × h, where l = length, b = breadth, and h = height.

Volume of a Cube

The space occupied by a cube. Formula: V = a³, where a is the length of an edge.

Capacity

The volume of a substance (usually a fluid) that a container can hold. It's often measured in litres (L) or millilitres (mL).

Important Unit Conversions

These are crucial for solving problems: 1 m³ = 1000 litres 1000 cm³ = 1 litre * 1 litre = 1000 mL

Step-by-Step Worked Examples

• Question: A cuboidal water tank is 6 m long, 5 m wide and 4.5 m deep. How many litres of water can it hold? (Given 1 m³ = 1000 L). Step 1: Identify the dimensions of the cuboid. Length (l) = 6 m Breadth (b) = 5 m Height/Depth (h) = 4.5 m Step 2: Calculate the volume of the tank in cubic meters. Volume (V) = l × b × h V = 6 m × 5 m × 4.5 m V = 30 × 4.5 m³ V = 135 m³ Step 3: Convert the volume from cubic meters to litres. We know that 1 m³ = 1000 litres. Capacity in litres = Volume in m³ × 1000 Capacity = 135 × 1000 L Capacity = 135,000 litres Final Answer: The tank can hold 135,000 litres of water.
• Question: A cuboidal pit 8 m long, 6 m wide and 3 m deep is dug. Find the cost of digging it at the rate of ₹30 per m³. Step 1: Calculate the volume of the cuboidal pit. This is the volume of earth that needs to be dug out. Length (l) = 8 m Breadth (b) = 6 m Depth (h) = 3 m Volume (V) = l × b × h V = 8 m × 6 m × 3 m V = 48 × 3 m³ V = 144 m³ Step 2: Calculate the total cost of digging. The rate is given as ₹30 per m³. Total Cost = Volume of pit × Rate per m³ Total Cost = 144 × 30 Total Cost = ₹ 4320 Final Answer: The total cost of digging the pit is ₹4320.

Exam Tips: Avoid These Common Mistakes

Two major errors often happen in volume problems. Be careful!

1. Ignoring Unit Consistency: Always check if the length, breadth, and height are in the same unit. If one dimension is in meters and another is in centimeters, you must convert them to a single unit before applying the formula V = l × b × h. Calculating with mixed units will always lead to a wrong answer.

1. Incorrect Litre Conversion: Students often get confused between the conversions for m³ and cm³. Remember this carefully:

• To convert from m³ to litres, you multiply by 1000. (Because 1 m³ is a large space holding 1000 L).
• To convert from cm³ to litres, you divide by 1000. (Because 1000 cm³ fit into a small 1 L container).

Practice Questions with Solutions

• Q: A matchbox measures 4 cm × 2.5 cm × 1.5 cm. What will be the volume of a packet containing 12 such boxes? A: Step 1: Calculate the volume of one matchbox. Given, l = 4 cm, b = 2.5 cm, h = 1.5 cm. Volume of one matchbox = l × b × h = 4 × 2.5 × 1.5 = 10 × 1.5 = 15 cm³. Step 2: Calculate the total volume of the packet containing 12 boxes. Total Volume = Volume of one box × Number of boxes Total Volume = 15 cm³ × 12 = 180 cm³. Final answer: The volume of the packet is 180 cm³.
• Q: A cuboidal vessel is 10 m long and 8 m wide. How high must it be made to hold 380 cubic metres of a liquid? A: Step 1: List the given information and the formula. Length (l) = 10 m Breadth (b) = 8 m Volume (V) = 380 m³ Height (h) = ? We know, Volume = l × b × h. Step 2: Substitute the values and solve for h. 380 = 10 × 8 × h 380 = 80 × h h = 380 / 80 h = 38 / 8 = 4.75 m. Final answer: The vessel must be 4.75 m high.
• Q: A godown measures 40 m × 25 m × 15 m. Find the maximum number of wooden crates each measuring 1.5 m × 1.25 m × 0.5 m that can be stored in the godown. A: Step 1: Calculate the volume of the godown. Volume of godown = 40 m × 25 m × 15 m = 15000 m³. Step 2: Calculate the volume of a single wooden crate. Volume of one crate = 1.5 m × 1.25 m × 0.5 m = 0.9375 m³. Step 3: Find the number of crates by dividing the godown's volume by the crate's volume. Number of crates = (Volume of godown) / (Volume of one crate) Number of crates = 15000 / 0.9375 = 16000. Final answer: A maximum of 16,000 wooden crates can be stored in the godown.
• Q: A river 3 m deep and 40 m wide is flowing at the rate of 2 km per hour. How much water will fall into the sea in a minute? A: Step 1: Convert the flow rate (speed) into meters per minute. This will act as the 'length' of the water flowing in one minute. Rate = 2 km/hr = 2000 m / 60 min = 100/3 m/min. So, the length of the water volume flowing in one minute is l = 100/3 m. Step 2: Identify the other dimensions. Depth (h) = 3 m Width (b) = 40 m Step 3: Calculate the volume of water flowing in one minute. Volume = l × b × h Volume = (100/3) m × 40 m × 3 m Volume = 100 × 40 × (3/3) m³ = 4000 m³. Final answer: 4000 m³ of water will fall into the sea in a minute.

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