NCERT Solutions for Class 9 Maths: Surface Area and Volume Ex 13.1

Understanding Surface Area of Cuboids and Cubes

Let's start with the basics. A cuboid is a 3D shape with six rectangular faces. Think of a shoebox. It has a length (l), a breadth (b), and a height (h).

• Total Surface Area (TSA) is the sum of the areas of all six faces. Since opposite faces are identical, we have two faces with area l × b (top and bottom), two with area b × h (sides), and two with area h × l (front and back). Summing them up gives us the formula:

TSA of a Cuboid = 2(lb + bh + hl)

• Lateral Surface Area (LSA) is the area of the four side faces only, excluding the top and bottom. This is often called the 'area of four walls'.

LSA of a Cuboid = 2(l + b)h

A cube is a special type of cuboid where all six faces are identical squares, meaning its length, breadth, and height are all equal. Let's call the side length 'a'.

• Total Surface Area (TSA): Since there are 6 identical square faces, each with an area of a²:

TSA of a Cube = 6a²

• Lateral Surface Area (LSA): This includes only the 4 side faces:

LSA of a Cube = 4a²

How to Derive the Cuboid's Surface Area Formula

1. Step 1: Visualize the 6 Faces — Imagine a cuboid (like a room). It has a floor, a ceiling, and four walls. These are its six rectangular faces. Let the dimensions be length (l), breadth (b), and height (h).
2. Step 2: Calculate Area of Paired Faces — Opposite faces of a cuboid are identical in area. - Area of the top face = Area of the bottom face = l × b - Area of the front face = Area of the back face = l × h - Area of the left side face = Area of the right side face = b × h
3. Step 3: Sum the Areas of All Faces — To find the Total Surface Area (TSA), we add the areas of all six faces together: TSA = (l × b) + (l × b) + (l × h) + (l × h) + (b × h) + (b × h)
4. Step 4: Simplify the Expression — Now, we group the like terms: TSA = 2(l × b) + 2(l × h) + 2(b × h) Factoring out the 2 gives us the final formula: TSA = 2(lb + bh + hl)

Step-by-Step Solved Examples (from Ex 13.1)

• Problem (Based on NCERT Ex 13.1, Q1): A plastic box 1.5 m long, 1.25 m wide and 65 cm deep is to be made. It is opened at the top. Find the area of the sheet required for making the box. Step 1: Unify the units. The length (l) = 1.5 m The breadth (b) = 1.25 m The depth (h) = 65 cm = 0.65 m (since 1 m = 100 cm). Step 2: Identify the required area. The box is open at the top. So, the required area is the area of the four walls (LSA) plus the area of the base. Area = LSA + Area of Base = [2(l + b)h] + [l × b] Step 3: Substitute the values and calculate. Area = [2(1.5 + 1.25) × 0.65] + [1.5 × 1.25] Area = [2(2.75) × 0.65] + [1.875] Area = [5.5 × 0.65] + 1.875 Area = 3.575 + 1.875 Area = 5.45 m² Final Answer: The area of the sheet required is 5.45 m².
• Problem (Based on NCERT Ex 13.1, Q2): The length, breadth and height of a room are 5 m, 4 m and 3 m respectively. Find the cost of white washing the walls of the room and the ceiling at the rate of ₹7.50 per m². Step 1: Identify the dimensions. Length (l) = 5 m Breadth (b) = 4 m Height (h) = 3 m Step 2: Determine the area to be whitewashed. We need to whitewash the four walls and the ceiling. The floor is not painted. Area to be whitewashed = (Area of four walls) + (Area of ceiling) Area = [2(l + b)h] + [l × b] Step 3: Calculate the area. Area = [2(5 + 4) × 3] + [5 × 4] Area = [2(9) × 3] + [20] Area = [18 × 3] + 20 Area = 54 + 20 Area = 74 m² Step 4: Calculate the total cost. Cost per m² = ₹7.50 Total cost = Total area × Rate Total cost = 74 × 7.50 = ₹555 Final Answer: The cost of whitewashing the walls and ceiling is ₹555.

Common Mistakes to Avoid

1. Ignoring Unit Conversion: This is the most common trap! If length is in meters and height is in centimeters, you MUST convert them to the same unit before using any formula. Always double-check units at the start of the problem.

2. LSA vs. TSA Confusion: Read the question very carefully. If it asks for the 'area of four walls', it means Lateral Surface Area (LSA). If it asks for the total area to cover a closed box, it's Total Surface Area (TSA).

3. Forgetting 'Open Top' Boxes: For questions involving a box that is open at the top (like a swimming pool or a carton without a lid), the area is NOT the TSA. You must calculate the area of the 4 walls (LSA) and add the area of the single base. The formula is: Area = 2(l + b)h + lb.

Practice Questions with Solutions

• Q: Find the Total Surface Area (TSA) and Lateral Surface Area (LSA) of a cube with an edge length of 8 cm. A: Step 1: Identify the given information. The edge of the cube (a) = 8 cm. Step 2: Use the formula for the TSA of a cube: TSA = 6a². TSA = 6 × (8)² = 6 × 64 = 384 cm². Step 3: Use the formula for the LSA of a cube: LSA = 4a². LSA = 4 × (8)² = 4 × 64 = 256 cm². Final answer: The TSA is 384 cm² and the LSA is 256 cm².
• Q: The floor of a rectangular hall has a perimeter of 250 m. If the cost of painting the four walls at the rate of ₹10 per m² is ₹15000, find the height of the hall. (Hint: Perimeter of floor = 2(l+b)) A: Step 1: Find the area of the four walls from the total cost and rate. Area of four walls (LSA) = Total Cost / Rate = 15000 / 10 = 1500 m². Step 2: Use the formula for LSA. We know LSA = 2(l + b)h. Step 3: The perimeter of the floor is given as 250 m. The formula for the perimeter of a rectangle is 2(l + b). So, 2(l + b) = 250 m. Step 4: Substitute the value of 2(l + b) into the LSA formula. 1500 = (250) × h h = 1500 / 250 = 6 m. Final answer: The height of the hall is 6 m.
• Q: Two cubes, each of volume 64 cm³, are joined end to end. Find the surface area of the resulting cuboid. A: Step 1: Find the side of each cube from its volume. Volume of a cube = a³. So, a³ = 64 cm³. Taking the cube root, a = 4 cm. Step 2: Visualize the new cuboid. When two cubes of side 4 cm are joined end to end, the new shape is a cuboid. The length (l) of the new cuboid = 4 cm + 4 cm = 8 cm. The breadth (b) remains the same = 4 cm. The height (h) remains the same = 4 cm. Step 3: Calculate the TSA of the resulting cuboid using the formula TSA = 2(lb + bh + hl). TSA = 2((8 × 4) + (4 × 4) + (4 × 8)) TSA = 2(32 + 16 + 32) TSA = 2(80) = 160 cm². Final answer: The surface area of the resulting cuboid is 160 cm².
• Q: Hameed has built a cubical water tank with a lid. The outer edge of the tank is 1.5 m long. He gets the outer surface of the tank excluding the base, covered with square tiles of side 25 cm. Find how much he would spend for the tiles, if the cost of the tiles is ₹360 per dozen. A: Step 1: Find the area to be tiled. The tank is cubical. It is covered on 5 faces (TSA - base area). Side of cube (a) = 1.5 m = 150 cm. Area to be tiled = 5a² = 5 × (150)² = 5 × 22500 = 112500 cm². Step 2: Find the area of one tile. Side of tile = 25 cm. Area of one tile = 25 × 25 = 625 cm². Step 3: Find the number of tiles required. Number of tiles = Total area to be tiled / Area of one tile = 112500 / 625 = 180 tiles. Step 4: Calculate the total cost. Cost is ₹360 per dozen (12 tiles). Number of dozens = 180 / 12 = 15 dozens. Total cost = Number of dozens × Cost per dozen = 15 × 360 = ₹5400. Final answer: Hameed would spend ₹5400 for the tiles.

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