Surface Area and Volume: Class 9 NCERT Chapter Guide

Understanding Key Terms and Shapes

Surface Area

The total area of the surface of a three-dimensional object. It's like the total area you would need to paint to cover the entire object. It is measured in square units (e.g., cm², m²).

Curved Surface Area (CSA) / Lateral Surface Area (LSA)

The area of only the curved faces of a 3D object, excluding the flat top and bottom bases. For a cuboid, it's the area of the four walls (LSA). For a cylinder, it's the area of the curved 'tube' part (CSA).

Total Surface Area (TSA)

The sum of the Curved/Lateral Surface Area and the area of the flat bases. For a cylinder, TSA = CSA + Area of top circle + Area of bottom circle.

Volume

The amount of space that a 3D object occupies. It represents the capacity of the object, like how much water a bottle can hold. It is measured in cubic units (e.g., cm³, m³).

Breaking Down the Formulas for Each Shape

Memorizing formulas is one thing, but understanding where they come from is the key to mastering this chapter. Let's break them down logically.

1. Cuboid and Cube:
A cuboid is a box with six rectangular faces. Its Total Surface Area (TSA) is simply the sum of the areas of these six faces: 2(lb + bh + hl). The Lateral Surface Area (LSA) is the area of the four 'walls', excluding the top and bottom, so it's 2(l+b)h. The Volume is the product of its three dimensions: l × b × h.
A cube is a special cuboid where all sides are equal (l=b=h=a). So, its TSA becomes 6a² and LSA is 4a². The volume is a³.

2. Right Circular Cylinder:
Imagine a cylinder is a stack of circles. The area of the base is πr². When you stack them up to a height 'h', the volume becomes πr²h. For the Curved Surface Area (CSA), imagine cutting the cylinder's side and unrolling it. You get a rectangle with height 'h' and length equal to the circumference of the base, 2πr. So, CSA = 2πrh. The Total Surface Area (TSA) is the CSA plus the area of the two circular bases: 2πrh + 2πr².

3. Right Circular Cone:
A cone's volume is exactly one-third of a cylinder with the same base radius and height: (1/3)πr²h. Its Curved Surface Area is πrl, where 'l' is the slant height. You can find 'l' using Pythagoras' theorem on the right-angled triangle formed by the radius, height, and slant height: l = √(r² + h²). The TSA is the CSA plus the area of its circular base: πrl + πr².

Step-by-Step Worked Examples

• Problem 1: Painting a Hall The length, breadth, and height of a hall are 20 m, 15 m, and 5 m respectively. Find the cost of painting the four walls and the ceiling at a rate of ₹20 per m². Solution: Step 1: Identify the areas to be painted. We need to paint the four walls (Lateral Surface Area) and the ceiling. The floor is not painted. Step 2: Calculate the Lateral Surface Area (Area of 4 walls). LSA of a cuboid = 2(l + b)h LSA = 2 (20 + 15) 5 LSA = 2 35 5 LSA = 350 m² Step 3: Calculate the area of the ceiling. Area of ceiling = l b Area = 20 15 Area = 300 m² Step 4: Calculate the total area to be painted. Total Area = LSA + Area of ceiling Total Area = 350 + 300 = 650 m² Step 5: Calculate the total cost. Cost = Total Area × Rate per m² Cost = 650 * 20 Cost = ₹13,000 Final Answer: The total cost of painting is ₹13,000.
• Problem 2: Capacity of a Conical Vessel The height of a cone is 21 cm and its slant height is 28 cm. Find the volume of the cone. (Use π = 22/7) Solution: Step 1: Identify the given values and what needs to be found. Height (h) = 21 cm Slant height (l) = 28 cm We need to find the Volume. The formula for volume is (1/3)πr²h. We don't have the radius (r) yet. Step 2: Calculate the radius (r) using the relationship between h, l, and r. We know l² = r² + h² (Pythagoras' theorem). 28² = r² + 21² 784 = r² + 441 r² = 784 - 441 r² = 343 So, r = √343 = 7√7 cm. Keeping it as r² is easier for the volume calculation. Step 3: Calculate the volume of the cone. Volume = (1/3) π r² h Volume = (1/3) (22/7) 343 21 Notice that 343 = 7 49, and we have a 7 in the denominator and a 21 in the numerator. Let's simplify. Volume = (1/3) (22/7) 343 21 Volume = 1 22 (343/7) (21/3) Volume = 22 49 7 Volume = 22 343 Volume = 7546 cm³ Final Answer: The volume of the cone is 7546 cm³.

Exam Tips & Common Mistakes to Avoid

1. Units are Crucial: Always check if all dimensions (length, radius, height) are in the same unit. If not, convert them before you start calculating. Your final answer's unit is also important: use square units (cm², m²) for area and cubic units (cm³, m³) for volume.

2. CSA vs. TSA: Read the question very carefully. Does it ask for the area of the 'four walls' (LSA)? Or to make a 'closed' container (TSA)? Or an 'open' container (LSA + area of one base)? Visualizing the situation helps avoid this common error.

3. Height (h) vs. Slant Height (l): This is a major confusion point for cones. The height (h) is the perpendicular distance from the vertex to the center of the base. The slant height (l) is the length of the sloping side. Always use Pythagoras' Theorem (l² = h² + r²) to relate them. Never use 'l' in the volume formula!

4. Hemisphere TSA: The curved surface area of a hemisphere is 2πr². But its total surface area is 3πr² because you must add the area of the flat circular top (πr²). Many students forget to add this top part for a solid hemisphere.

Practice Questions with Solutions

• Q: The curved surface area of a right circular cylinder of height 14 cm is 88 cm². Find the diameter of the base of the cylinder. A: Step 1: Write down the formula for the Curved Surface Area (CSA) of a cylinder: CSA = 2πrh. Step 2: Substitute the given values into the formula. We are given CSA = 88 cm², h = 14 cm. So, 88 = 2 (22/7) r 14. Step 3: Solve for the radius (r). 88 = 2 22 r (14/7) => 88 = 44 r 2 => 88 = 88 r. Therefore, r = 1 cm. Step 4: Calculate the diameter. Diameter = 2 radius = 2 * 1 = 2 cm. Final answer: The diameter of the base of the cylinder is 2 cm.
• Q: A cubical box has each edge 10 cm and another cuboidal box is 12.5 cm long, 10 cm wide and 8 cm high. Which box has the greater lateral surface area and by how much? A: Step 1: Calculate the Lateral Surface Area (LSA) of the cubical box. LSA of cube = 4a² where a = 10 cm. LSA = 4 (10)² = 4 100 = 400 cm². Step 2: Calculate the Lateral Surface Area (LSA) of the cuboidal box. LSA of cuboid = 2(l+b)h. Here, l=12.5 cm, b=10 cm, h=8 cm. LSA = 2 (12.5 + 10) 8 = 2 22.5 8 = 360 cm². Step 3: Compare the two LSAs. LSA of cube (400 cm²) > LSA of cuboid (360 cm²). Step 4: Find the difference. Difference = 400 - 360 = 40 cm². Final answer: The cubical box has the greater lateral surface area by 40 cm².
• Q: Find the volume of a sphere whose surface area is 154 cm². (Use π = 22/7) A: Step 1: Use the surface area formula of a sphere to find the radius. Surface Area = 4πr². We are given Surface Area = 154 cm². So, 154 = 4 (22/7) r². Step 2: Solve for r². 154 = (88/7) r² => r² = (154 7) / 88 = (7 7) / 4. So, r = √(49/4) = 7/2 = 3.5 cm. Step 3: Now, use the radius to find the volume of the sphere. Volume = (4/3)πr³. Volume = (4/3) (22/7) (7/2)³. Step 4: Calculate the final volume. Volume = (4/3) (22/7) (343/8) = (4 22 343) / (3 7 8) = (1 11 * 49) / 3 = 539 / 3 cm³. Final answer: The volume of the sphere is 539/3 cm³ (or approx 179.67 cm³).
• Q: The inner diameter of a cylindrical wooden pipe is 24 cm and its outer diameter is 28 cm. The length of the pipe is 35 cm. Find the mass of the pipe, if 1 cm³ of wood has a mass of 0.6 g. A: Step 1: Find the inner radius (r) and outer radius (R). Inner radius r = 24/2 = 12 cm. Outer radius R = 28/2 = 14 cm. Height (h) = 35 cm. Step 2: Calculate the volume of the wood. The volume of wood is the difference between the outer volume and the inner volume of the cylinder. Volume = πR²h - πr²h = πh(R² - r²). Step 3: Substitute the values. Volume = (22/7) 35 (14² - 12²) = 22 5 (196 - 144) = 110 52 = 5720 cm³. Step 4: Calculate the total mass of the pipe. Mass = Volume of wood × mass per cm³. Mass = 5720 0.6 = 3432 g. Final answer: The mass of the pipe is 3432 g or 3.432 kg.

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