CBSE Class 9 Maths: Surface Area and Volume - Exercise 13.6 (Volume of Cylinders)

Understanding the Volume of a Cylinder

A cylinder is a three-dimensional solid that holds a special place in geometry due to its perfectly round base and uniform height. Imagine stacking many identical circles one on top of the other; that's essentially what a cylinder is. The space occupied by such a solid is called its volume.

To calculate the volume of a cylinder, we need two key measurements: the radius (r) of its circular base and its height (h). The radius is the distance from the center of the circular base to any point on its circumference, and the height is the perpendicular distance between its two circular bases.

The fundamental idea behind calculating the volume of any prism-like solid (which a cylinder effectively is, with a circular base) is to multiply the area of its base by its height. For a cylinder, the base is a circle, and the area of a circle is given by the formula $\pi r^2$. Therefore, the volume of a cylinder is:

Formula for the Volume of a Cylinder:
$V = \pi r^2 h$
Where:

• $V$ represents the Volume of the cylinder.
• $\pi$ (Pi) is a mathematical constant, approximately $\frac{22}{7}$ or $3.14$.
• $r$ is the radius of the circular base.
• $h$ is the height of the cylinder.

The unit of volume is always in cubic units, such as cubic centimetres ($cm^3$) or cubic metres ($m^3$), because it represents a three-dimensional space. Remember to always ensure that the units for radius and height are consistent before applying the formula. For instance, if the radius is in cm and height in m, convert one to match the other.

Solving Problems: Step-by-Step Examples

• Example 1: Finding Volume with Given Dimensions _Question:_ A cylindrical vessel has a base radius of 7 cm and a height of 10 cm. Find the volume of water it can hold. _Solution:_ Step 1: Identify the given values. Radius ($r$) = 7 cm Height ($h$) = 10 cm Constant $\pi = \frac{22}{7}$ Step 2: Apply the formula for the volume of a cylinder. $V = \pi r^2 h$ Step 3: Substitute the values and calculate. $V = \frac{22}{7} \times (7 \text{ cm})^2 \times 10 \text{ cm}$ $V = \frac{22}{7} \times 49 \text{ cm}^2 \times 10 \text{ cm}$ $V = 22 \times 7 \text{ cm}^2 \times 10 \text{ cm}$ $V = 154 \times 10 \text{ cm}^3$ $V = 1540 \text{ cm}^3$ Final Answer: The volume of water the vessel can hold is 1540 cubic centimetres.
• Example 2: Finding Height Given Volume and Radius _Question:_ The volume of a cylindrical tank is 3080 $m^3$. If its base radius is 7 m, find the height of the tank. _Solution:_ Step 1: Identify the given values. Volume ($V$) = 3080 $m^3$ Radius ($r$) = 7 m Constant $\pi = \frac{22}{7}$ Step 2: Start with the volume formula and rearrange to find height. $V = \pi r^2 h$ $h = \frac{V}{\pi r^2}$ Step 3: Substitute the values and calculate. $h = \frac{3080 \text{ m}^3}{\frac{22}{7} \times (7 \text{ m})^2}$ $h = \frac{3080}{\frac{22}{7} \times 49}$ $h = \frac{3080}{22 \times 7}$ $h = \frac{3080}{154}$ $h = 20 \text{ m}$ Final Answer: The height of the cylindrical tank is 20 metres.
• Example 3: Handling Diameter and Unit Conversion _Question:_ A soft drink is available in two packs: (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm, and (ii) a plastic cylinder with a circular base of diameter 7 cm and height 10 cm. Which pack has greater capacity and by how much? _Solution:_ Step 1: Calculate the volume of the tin can (cuboid). Length ($l$) = 5 cm, Width ($w$) = 4 cm, Height ($h_1$) = 15 cm Volume of cuboid ($V_1$) = $l \times w \times h_1 = 5 \times 4 \times 15 = 300 \text{ cm}^3$ Step 2: Calculate the volume of the plastic cylinder. Diameter = 7 cm, so Radius ($r$) = $\frac{\text{Diameter}}{2} = \frac{7}{2}$ cm Height ($h_2$) = 10 cm Volume of cylinder ($V_2$) = $\pi r^2 h_2 = \frac{22}{7} \times (\frac{7}{2})^2 \times 10$ $V_2 = \frac{22}{7} \times \frac{49}{4} \times 10$ $V_2 = \frac{22 \times 7}{4} \times 10 = \frac{154}{4} \times 10 = 38.5 \times 10 = 385 \text{ cm}^3$ Step 3: Compare the volumes and find the difference. $V_2 = 385 \text{ cm}^3$, $V_1 = 300 \text{ cm}^3$ Since $385 > 300$, the cylindrical pack has greater capacity. Difference = $V_2 - V_1 = 385 - 300 = 85 \text{ cm}^3$ Final Answer: The cylindrical pack has a greater capacity by 85 cubic centimetres.

Smart Tips for Exam Success

To ace problems related to the volume of cylinders, keep these crucial tips in mind:

1. Diameter vs. Radius: Always check if the problem provides the diameter or the radius. If diameter is given, remember to divide it by 2 to get the radius ($r = \frac{D}{2}$) before using the formula. This is a very common mistake!
2. Unit Consistency: Ensure all dimensions (radius and height) are in the same units before calculating. If one is in cm and another in m, convert one to match the other. For example, 1 m = 100 cm. The final volume unit will be consistent (e.g., $cm^3$ or $m^3$).
3. Value of Pi ($\pi$): Unless specified, use $\pi = \frac{22}{7}$. If the problem states to use $\pi = 3.14$, then strictly adhere to that value. Using the fraction often simplifies calculations, especially when dimensions are multiples of 7.
4. Careful Calculations: Double-check your arithmetic, especially squaring the radius ($r^2$) and multiplication. A small error can lead to a completely wrong answer.
5. Formula Recall: Memorize the formula $V = \pi r^2 h$. Understanding its derivation (base area $\times$ height) helps in remembering it correctly.

Practice Questions with Solutions

• Q: A cylindrical pillar is 50 cm in diameter and 3.5 m in height. Find the volume of the pillar. (Use $\pi = \frac{22}{7}$) A: Step 1: Convert units to be consistent. Diameter = 50 cm, so Radius ($r$) = 25 cm = 0.25 m. Height ($h$) = 3.5 m. Step 2: Apply the volume formula. $V = \pi r^2 h$ Step 3: Substitute values and calculate. $V = \frac{22}{7} \times (0.25)^2 \times 3.5$ $V = \frac{22}{7} \times 0.0625 \times 3.5$ $V = 22 \times 0.0625 \times 0.5$ (since $3.5/7 = 0.5$) $V = 22 \times 0.03125$ $V = 0.6875 \text{ m}^3$ Final answer: The volume of the pillar is 0.6875 cubic metres.
• Q: The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold? (1000 $cm^3$ = 1 Litre) A: Step 1: Use the circumference to find the radius. Circumference ($C$) = $2\pi r = 132 \text{ cm}$ $2 \times \frac{22}{7} \times r = 132$ $r = \frac{132 \times 7}{2 \times 22} = \frac{66 \times 7}{22} = 3 \times 7 = 21 \text{ cm}$ Height ($h$) = 25 cm. Step 2: Calculate the volume in $cm^3$. $V = \pi r^2 h = \frac{22}{7} \times (21)^2 \times 25$ $V = \frac{22}{7} \times 441 \times 25$ $V = 22 \times 63 \times 25$ $V = 34650 \text{ cm}^3$ Step 3: Convert volume from $cm^3$ to litres. $1000 \text{ cm}^3 = 1 \text{ Litre}$ Volume in litres = $\frac{34650}{1000} = 34.65 \text{ Litres}$ Final answer: The cylindrical vessel can hold 34.65 litres of water.
• Q: A patient in a hospital is given soup daily in a cylindrical bowl of diameter 7 cm. If the bowl is filled with soup to a height of 4 cm, how much soup does the hospital have to prepare daily to serve 250 patients? A: Step 1: Find the radius of the bowl. Diameter = 7 cm, so Radius ($r$) = $\frac{7}{2}$ cm = 3.5 cm. Height of soup ($h$) = 4 cm. Step 2: Calculate the volume of soup in one bowl. $V_{\text{one bowl}} = \pi r^2 h = \frac{22}{7} \times (3.5)^2 \times 4$ $V_{\text{one bowl}} = \frac{22}{7} \times 12.25 \times 4$ $V_{\text{one bowl}} = 22 \times 1.75 \times 4$ (since $12.25/7 = 1.75$) $V_{\text{one bowl}} = 22 \times 7 = 154 \text{ cm}^3$ Step 3: Calculate the total volume of soup needed for 250 patients. Total Volume = $V_{\text{one bowl}} \times 250 = 154 \times 250$ Total Volume = 38500 $cm^3$ Final answer: The hospital has to prepare 38500 cubic centimetres of soup daily.
• Q: A closed cylindrical tank has a height of 1 m and a base diameter of 140 cm. What is its capacity in cubic metres? A: Step 1: Convert units to be consistent. Diameter = 140 cm, so Radius ($r$) = 70 cm = 0.7 m. Height ($h$) = 1 m. Step 2: Apply the volume formula. $V = \pi r^2 h$ Step 3: Substitute values and calculate. $V = \frac{22}{7} \times (0.7)^2 \times 1$ $V = \frac{22}{7} \times 0.49 \times 1$ $V = 22 \times 0.07 \times 1$ $V = 1.54 \text{ m}^3$ Final answer: The capacity of the cylindrical tank is 1.54 cubic metres.
• Q: The capacity of a closed cylindrical vessel of height 1 m is 15.4 litres. Find the radius of the base. (1 Litre = 1000 $cm^3$, Use $\pi = 3.14$ for this problem) A: Step 1: Convert units to be consistent. It's easier to work in cm and convert at the end, or ensure all are in m from the start. Height ($h$) = 1 m = 100 cm. Capacity ($V$) = 15.4 litres = $15.4 \times 1000 \text{ cm}^3 = 15400 \text{ cm}^3$. Step 2: Apply the volume formula and rearrange to find radius. $V = \pi r^2 h$ $r^2 = \frac{V}{\pi h}$ Step 3: Substitute values and calculate. $r^2 = \frac{15400}{3.14 \times 100}$ $r^2 = \frac{15400}{314}$ $r^2 = 49.044... \approx 49$ $r = \sqrt{49} = 7 \text{ cm}$ Final answer: The radius of the base of the cylindrical vessel is 7 centimetres.

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