NCERT Solutions Class 8 Maths Chapter 11 Mensuration Ex 11.4

Understanding Volume, Capacity, and the Core Formulas

Before solving the problems in class 8 maths mensuration ex 11 4, it is vital to understand the difference between volume and capacity.

• Volume refers to the amount of three-dimensional space occupied by an object. It is measured in cubic units, such as $cm^3$ or $m^3$.
• Capacity refers to the volume of substance (usually liquid or gas) that a container can hold. For example, a bottle has a physical volume (the space its plastic shell takes up on a desk) and a capacity (the amount of water it can hold inside).

Here are the fundamental formulas you must memorize:

1. Volume of a Cuboid = $Length \times Breadth \times Height = l \times b \times h$
2. Volume of a Cube = $Side^3 = a^3$
3. Volume of a Cylinder = $\pi r^2 h$ (where $r$ is the radius of the circular base and $h$ is the height of the cylinder)

Unit Conversions to Remember:

• $1 \text{ cm}^3 = 1 \text{ mL}$
• $1000 \text{ cm}^3 = 1 \text{ litre}$
• $1 \text{ m}^3 = 1000 \text{ litres} = 1,000,000 \text{ cm}^3$

Step-by-Step Approach to Solve Exercise 11.4 Problems

1. Identify the 3D Shape — Read the word problem carefully to identify if the object is a cube, a cuboid, or a cylinder.
2. Check and Standardize Units — Ensure all given dimensions (length, radius, height) are in the same unit. If some are in meters and others in centimeters, convert them first before applying formulas.
3. Apply the Correct Formula — Use the respective formula for volume. Remember that if the base area of a cuboid is given, volume can also be written as Base Area multiplied by Height.
4. Convert Units for Capacity — If the final answer asks for fluid volume (like litres), use the standard conversion factors ($1 \text{ m}^3 = 1000 \text{ L}$).

Crucial Tips and Pitfalls to Avoid in Mensuration

Many students lose marks on simple calculation errors. Keep these expert tips in mind:

• Diameter vs. Radius: Frequently, questions provide the diameter of a cylindrical tank rather than its radius. Always divide the diameter by 2 to find the radius ($r = d/2$) before substituting it into $\pi r^2 h$.
• Unit Consistency: Mixing meters and centimeters is a common trap. If a cuboid is $2\text{ m}$ long, $1\text{ m}$ wide, and $50\text{ cm}$ high, convert $50\text{ cm}$ to $0.5\text{ m}$ before multiplying.
• Volume vs. Surface Area: Do not confuse outer area with inner storage. If a question asks how much paint is needed for a tank, calculate surface area. If it asks how much water a tank can hold, calculate volume.

Practice Questions with Solutions

• Q: 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 $1\text{ m}^3 = 1000$ litres) A: Step 1: Identify the given dimensions of the cuboidal tank. Length ($l$) = $6 \text{ m}$ Breadth ($b$) = $5 \text{ m}$ Height/Depth ($h$) = $4.5 \text{ m}$ Step 2: Apply the formula for the volume of a cuboid. Volume ($V$) = $l \times b \times h$ $V = 6 \times 5 \times 4.5 = 30 \times 4.5 = 135 \text{ m}^3$ Step 3: Convert the volume from cubic meters into liters. Since $1 \text{ m}^3 = 1000$ litres: Capacity = $135 \times 1000 = 135,000$ litres. Final answer: The water tank can hold 135,000 litres of water.
• Q: Find the height of a cuboid whose base area is $180 \text{ cm}^2$ and volume is $900 \text{ cm}^3$. A: Step 1: Recall the formula relating volume, base area, and height. Volume of a cuboid = $\text{Length} \times \text{Breadth} \times \text{Height}$ Since $\text{Base Area} = \text{Length} \times \text{Breadth}$, we can write: $\text{Volume} = \text{Base Area} \times \text{Height}$ Step 2: Plug in the given values into the formula. Given, $\text{Volume} = 900 \text{ cm}^3$ and $\text{Base Area} = 180 \text{ cm}^2$. $900 = 180 \times \text{Height}$ Step 3: Solve for the height. $\text{Height} = 900 / 180 = 5 \text{ cm}$ Final answer: The height of the cuboid is 5 cm.
• Q: A cuboid has dimensions $60 \text{ cm} \times 54 \text{ cm} \times 30 \text{ cm}$. How many small cubes with side $6 \text{ cm}$ can be placed inside this cuboid? A: Step 1: Calculate the volume of the large cuboid. Volume of cuboid = $l \times b \times h = 60 \times 54 \times 30 = 97,200 \text{ cm}^3$ Step 2: Calculate the volume of one small cube. Volume of a cube = $\text{side}^3 = 6 \times 6 \times 6 = 216 \text{ cm}^3$ Step 3: Find the total number of cubes that can fit. Number of cubes = $\text{Volume of cuboid} / \text{Volume of one cube}$ Number of cubes = $97,200 / 216 = 450$ Final answer: 450 small cubes can be placed in the given cuboid.
• Q: A milk tank is in the form of a cylinder whose radius is $1.5 \text{ m}$ and length (height) is $7 \text{ m}$. Find the quantity of milk in litres that can be stored in the tank. (Take $\pi = 22/7$) A: Step 1: Identify the given dimensions of the cylindrical tank. Radius ($r$) = $1.5 \text{ m}$ Height/Length ($h$) = $7 \text{ m}$ Step 2: Apply the formula for the volume of a cylinder. Volume ($V$) = $\pi r^2 h = (22/7) \times (1.5)^2 \times 7$ $V = 22 \times 2.25 = 49.5 \text{ m}^3$ Step 3: Convert the volume into litres. Since $1 \text{ m}^3 = 1000$ litres: Quantity of milk = $49.5 \times 1000 = 49,500$ litres. Final answer: The quantity of milk that can be stored in the tank is 49,500 litres.

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