CBSE Class 9 Maths: Surface Area and Volume - Exercise 13.2 (Cylinder)

Understanding Surface Area of a Cylinder

A cylinder is a three-dimensional solid shape with two identical circular bases connected by a curved surface. Think of everyday objects like a soda can, a water pipe, or a drum. When we talk about the surface area of a cylinder, we are referring to the total area of all its surfaces that you can touch or paint. This includes the area of its two circular bases (top and bottom) and the area of its curved side.

The Curved Surface Area (CSA) of a cylinder is the area of just its curved part, excluding the top and bottom circular bases. Imagine unwrapping the label from a can – that's the curved surface. If you unroll this curved surface, it forms a perfect rectangle. The length of this rectangle is equal to the circumference of the circular base (which is $2\pi r$), and its width is the height of the cylinder (denoted by $h$). Therefore, the formula for the Curved Surface Area (CSA) of a cylinder is $2\pi rh$, where '$r

The Total Surface Area (TSA) of a cylinder includes the curved surface area plus the area of both circular bases. The area of one circular base is $\pi r^2$. Since there are two bases (top and bottom), their combined area is $2\pi r^2$. Adding this to the CSA, we get the formula for the Total Surface Area (TSA) of a cylinder: $2\pi rh + 2\pi r^2$. This can also be written in a more factored form as $2\pi r(h + r)$ by taking $2\pi r$ common. Understanding these formulas and when to apply them based on the problem statement is fundamental to solving problems in Exercise 13.2.

Key Definitions for Cylinders

Cylinder

A three-dimensional solid geometric shape that has two parallel circular bases and a curved surface connecting them.

Radius (r)

The distance from the center to any point on the circumference of the circular base of the cylinder.

Height (h)

The perpendicular distance between the two parallel circular bases of the cylinder.

Curved Surface Area (CSA)

The area of the curved lateral surface of the cylinder, excluding the areas of the top and bottom circular bases. Its formula is $2\pi rh$.

Total Surface Area (TSA)

The sum of the curved surface area and the areas of the two circular bases. Its formula is $2\pi r(h + r)$.

Step-by-Step Worked Examples

• Example 1: Finding Curved Surface Area A cylindrical pillar has a diameter of 50 cm and a height of 3.5 m. Find the cost of painting the curved surface of the pillar at the rate of ₹12.50 per square meter. Step 1: Identify given values and ensure consistent units. Diameter (d) = 50 cm. So, Radius (r) = d/2 = 50/2 = 25 cm. Height (h) = 3.5 m. Since the cost is per square meter, convert radius to meters: r = 25 cm = 0.25 m. Rate of painting = ₹12.50 per m². Step 2: Apply the formula for Curved Surface Area (CSA). CSA of a cylinder = $2\pi rh$ CSA = $2 \times \frac{22}{7} \times 0.25 \times 3.5$ CSA = $2 \times \frac{22}{7} \times \frac{1}{4} \times \frac{7}{2}$ (converting decimals to fractions for easier calculation) CSA = $\frac{2 \times 22 \times 1 \times 7}{7 \times 4 \times 2}$ CSA = $\frac{308}{56}$ = $5.5$ m². Step 3: Calculate the total cost. Cost of painting = CSA $\times$ Rate Cost = $5.5 \times 12.50$ Cost = ₹68.75. Final Answer: The cost of painting the curved surface of the pillar is ₹68.75.
• Example 2: Finding Total Surface Area A closed cylindrical tank of radius 7 m and height 3 m is made from a sheet of metal. How much sheet metal is required? Step 1: Identify given values. Radius (r) = 7 m Height (h) = 3 m Since the tank is closed, we need to find the Total Surface Area (TSA). Step 2: Apply the formula for Total Surface Area (TSA). TSA of a cylinder = $2\pi r(h + r)$ TSA = $2 \times \frac{22}{7} \times 7 \times (3 + 7)$ TSA = $2 \times \frac{22}{7} \times 7 \times 10$ TSA = $2 \times 22 \times 10$ (The 7's cancel out) TSA = $44 \times 10$ TSA = $440$ m². Final Answer: $440$ m² of sheet metal is required to make the closed cylindrical tank.
• Example 3: Finding Dimensions Given Surface Area The curved surface area of a right circular cylinder is $4.4 \text{ m}^2$. If the radius of the base of the cylinder is $0.7 \text{ m}$, find its height. Step 1: Identify given values. Curved Surface Area (CSA) = $4.4 \text{ m}^2$ Radius (r) = $0.7 \text{ m}$ We need to find the height (h). Step 2: Use the CSA formula and substitute known values. CSA = $2\pi rh$ $4.4 = 2 \times \frac{22}{7} \times 0.7 \times h$ Step 3: Solve for h. $4.4 = 2 \times \frac{22}{7} \times \frac{7}{10} \times h$ $4.4 = 2 \times \frac{22}{10} \times h$ $4.4 = \frac{44}{10} \times h$ $4.4 = 4.4 \times h$ $h = \frac{4.4}{4.4}$ $h = 1 \text{ m}$. Final Answer: The height of the cylinder is 1 meter.

Essential Exam Tips for Surface Area and Volume

Always pay close attention to the units given in the problem and ensure consistency. If the radius is in cm and height in m, convert one to match the other before calculation. For example, convert 1 m to 100 cm or 25 cm to 0.25 m. Also, read carefully whether the question asks for Curved Surface Area (CSA) or Total Surface Area (TSA). A question asking for "the area to be painted on the outside of a pipe" usually implies CSA, while "the amount of material needed to make a closed cylindrical container" implies TSA. If a container is open at one end, remember to calculate CSA + area of one base. Remember to use the value of $\pi$ as $22/7$ or $3.14$, as specified, or $22/7$ if not specified. Double-check your calculations, especially with squares and multiplications, to avoid silly errors. Visualize the shape and what each part of the formula represents; this helps in understanding and retaining the concepts.

Practice Questions with Solutions

• Q: The diameter of a road roller is 84 cm and its length is 1 m. If it takes 750 complete revolutions to move once over a level road, find the area of the road. A: Step 1: Identify given dimensions and convert to consistent units. Diameter = 84 cm, so Radius (r) = 84/2 = 42 cm = 0.42 m. Length (h) = 1 m. Step 2: Calculate the area covered in one revolution (Curved Surface Area). Area covered in one revolution = CSA = $2\pi rh$ CSA = $2 \times \frac{22}{7} \times 0.42 \times 1$ CSA = $2 \times 22 \times 0.06 \times 1 = 2.64 \text{ m}^2$. Step 3: Calculate the total area of the road. Total area = Area covered in one revolution $\times$ Number of revolutions Total area = $2.64 \times 750 = 1980 \text{ m}^2$. Final answer: The area of the road is $1980 \text{ m}^2$.
• Q: A cylindrical vessel, open from the top, has a base radius of 14 cm and height of 20 cm. Find the cost of painting its inner and outer curved surface and its base at the rate of ₹5 per 100 cm². A: Step 1: Identify given dimensions. Radius (r) = 14 cm, Height (h) = 20 cm. Step 2: Calculate the area to be painted. This includes inner CSA, outer CSA, and the base area. Area to be painted = (Inner CSA + Outer CSA) + Base Area Inner CSA = $2\pi rh = 2 \times \frac{22}{7} \times 14 \times 20 = 2 \times 22 \times 2 \times 20 = 1760 \text{ cm}^2$. Outer CSA is also $1760 \text{ cm}^2$ (assuming negligible thickness). Base Area = $\pi r^2 = \frac{22}{7} \times 14 \times 14 = 22 \times 2 \times 14 = 616 \text{ cm}^2$. Total Area = $1760 + 1760 + 616 = 4136 \text{ cm}^2$. Step 3: Calculate the cost of painting. Rate = ₹5 per 100 cm² = ₹0.05 per cm². Cost = Total Area $\times$ Rate per cm² Cost = $4136 \times 0.05 = ₹206.80$. Final answer: The cost of painting is ₹206.80.
• Q: It is required to make a closed cylindrical tank of height 1 m and base diameter 140 cm from a metal sheet. How many square meters of the sheet are required for the same? A: Step 1: Identify given dimensions and convert to consistent units. Height (h) = 1 m. Base Diameter = 140 cm, so Radius (r) = 140/2 = 70 cm = 0.7 m. Step 2: Calculate the Total Surface Area (TSA) as it's a closed tank. TSA = $2\pi r(h + r)$ TSA = $2 \times \frac{22}{7} \times 0.7 \times (1 + 0.7)$ TSA = $2 \times \frac{22}{7} \times \frac{7}{10} \times 1.7$ TSA = $2 \times \frac{22}{10} \times 1.7 = 4.4 \times 1.7 = 7.48 \text{ m}^2$. Final answer: $7.48 \text{ m}^2$ of metal sheet is required.
• 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 cm³ = 1 litre) A: Step 1: Identify given values. Circumference of base = $2\pi r = 132 \text{ cm}$. Height (h) = 25 cm. We need to find the volume, which requires radius. Step 2: Find the radius (r) from the circumference. $2\pi r = 132$ $2 \times \frac{22}{7} \times r = 132$ $r = \frac{132 \times 7}{2 \times 22} = \frac{132 \times 7}{44} = 3 \times 7 = 21 \text{ cm}$. Step 3: Calculate the volume of the cylinder. Volume (V) = $\pi r^2 h$ $V = \frac{22}{7} \times 21 \times 21 \times 25$ $V = 22 \times 3 \times 21 \times 25 = 66 \times 525 = 34650 \text{ cm}^3$. Step 4: Convert volume from cm³ to litres. Since 1000 cm³ = 1 litre, $34650 \text{ cm}^3 = \frac{34650}{1000} = 34.65 \text{ litres}$. Final answer: The cylindrical vessel can hold $34.65$ litres of water.
• Q: A metal pipe is 77 cm long. The inner diameter of a cross section is 4 cm, the outer diameter is 4.4 cm. Find its (i) inner curved surface area, (ii) outer curved surface area, (iii) total surface area. A: Step 1: Identify given dimensions and radii. Length (height, h) = 77 cm. Inner diameter = 4 cm, so inner radius ($r_1$) = 2 cm. Outer diameter = 4.4 cm, so outer radius ($r_2$) = 2.2 cm. Step 2: Calculate inner curved surface area. Inner CSA = $2\pi r_1 h = 2 \times \frac{22}{7} \times 2 \times 77 = 2 \times 22 \times 2 \times 11 = 968 \text{ cm}^2$. Step 3: Calculate outer curved surface area. Outer CSA = $2\pi r_2 h = 2 \times \frac{22}{7} \times 2.2 \times 77 = 2 \times \frac{22}{7} \times \frac{22}{10} \times 77 = 2 \times 22 \times \frac{22}{10} \times 11 = \frac{10648}{10} = 1064.8 \text{ cm}^2$. Step 4: Calculate total surface area (includes inner CSA, outer CSA, and areas of the two circular rings at the ends). Area of one end ring = $\pi (r_2^2 - r_1^2) = \frac{22}{7} (2.2^2 - 2^2) = \frac{22}{7} (4.84 - 4) = \frac{22}{7} \times 0.84 = 22 \times 0.12 = 2.64 \text{ cm}^2$. Area of two end rings = $2 \times 2.64 = 5.28 \text{ cm}^2$. Total Surface Area = Inner CSA + Outer CSA + Area of two end rings Total Surface Area = $968 + 1064.8 + 5.28 = 2038.08 \text{ cm}^2$. Final answer: (i) Inner CSA = $968 \text{ cm}^2$, (ii) Outer CSA = $1064.8 \text{ cm}^2$, (iii) Total Surface Area = $2038.08 \text{ cm}^2$.

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