Mensuration Exercise 11.3: Surface Area of 3D Shapes (Class 8 Maths)

Welcome to a deep dive into Mensuration Exercise 11.3 for Class 8 Maths! This chapter is all about understanding and calculating the surface area of common three-dimensional shapes like cuboids, cubes, and cylinders. Think about wrapping a gift, painting a room, or covering a cylindrical tank – all these real-world tasks require you to know the surface area.

Here, you'll not only learn the essential formulas but also understand when to use each one, distinguishing between Total Surface Area (TSA) and Lateral/Curved Surface Area (LSA/CSA). We'll go through step-by-step examples and tackle common tricky questions. By the end of this page, you'll be confident in solving any problem related to surface areas of these shapes, building a strong foundation for your geometry journey. Let's get started and master these vital concepts!

What is Surface Area? (Cuboids, Cubes, Cylinders)

Imagine you have a shoebox. If you wanted to cover its entire outer surface with decorative paper, the amount of paper needed would be its Total Surface Area (TSA). In simple terms, the surface area of a 3D object is the sum of the areas of all its outer faces or surfaces. It's like flattening out the 3D shape and measuring the total area of that flattened shape. Surface area is always measured in square units, like cm² or m².

For shapes like cuboids and cubes, TSA is the sum of the areas of all six faces. A Lateral Surface Area (LSA), sometimes called the area of four walls for cuboids/cubes, refers only to the area of the sides, excluding the top and bottom faces. This is useful when painting the walls of a room but not the floor or ceiling.

For a cylinder, which has a curved surface and two circular bases, we talk about Curved Surface Area (CSA) for just the curved part. The Total Surface Area (TSA) of a cylinder includes the CSA plus the areas of the two circular bases. Understanding this distinction is crucial for solving problems correctly.

Essential Formulas for Surface Area

Total Surface Area (TSA) of a Cuboid
The sum of the areas of all six rectangular faces. Formula: 2(lb + bh + hl), where l = length, b = breadth, h = height.
Lateral Surface Area (LSA) of a Cuboid
The sum of the areas of the four vertical faces (excluding top and bottom). Formula: 2(l + b)h.
Total Surface Area (TSA) of a Cube
The sum of the areas of all six square faces. Formula: 6a², where a = side length of the cube.
Lateral Surface Area (LSA) of a Cube
The sum of the areas of the four vertical faces. Formula: 4a².
Curved Surface Area (CSA) of a Cylinder
The area of the curved part of the cylinder (excluding top and bottom circles). Formula: 2πrh, where r = radius of base, h = height.
Total Surface Area (TSA) of a Cylinder
The sum of the curved surface area and the areas of the two circular bases. Formula: 2πr(h + r).

Step-by-Step Calculation of Surface Area

  1. Step 1: Identify the 3D Shape and Goal — First, carefully read the problem to determine if the object is a cuboid, cube, or cylinder. Then, identify whether you need to calculate the Total Surface Area (TSA) or the Lateral/Curved Surface Area (LSA/CSA). Pay attention to keywords like 'area of four walls', 'open tank', or 'entire surface'.
  2. Step 2: Note Down Dimensions — Write down all the given dimensions (length, breadth, height for cuboids; side for cubes; radius and height for cylinders). Make sure all units are consistent (e.g., all in cm or all in m). If not, convert them.
  3. Step 3: Choose the Correct Formula — Based on the shape and what you need to calculate (TSA or LSA/CSA), select the appropriate formula from the list above. For example, if it's a cuboid and you need the area of four walls, use 2(l+b)h.
  4. Step 4: Substitute Values and Calculate — Substitute the numerical values of the dimensions into the chosen formula. Perform the calculations carefully, paying attention to order of operations (PEMDAS/BODMAS). Use the value of π as 22/7 or 3.14, as specified or appropriate.
  5. Step 5: State the Final Answer with Units — After calculating, write down your final answer with the correct square units (e.g., cm², m²). Double-check your calculation if time permits.

Worked Examples: Applying Surface Area Formulas

  • Example 1: Cuboid's Total Surface Area A cuboidal box has length 10 cm, breadth 8 cm, and height 6 cm. Find its total surface area. Solution: Step 1: Identify the shape as a cuboid. We need to find the Total Surface Area (TSA). Step 2: Given dimensions: l = 10 cm, b = 8 cm, h = 6 cm. Step 3: The formula for TSA of a cuboid is 2(lb + bh + hl). Step 4: Substitute the values: TSA = 2((10 × 8) + (8 × 6) + (6 × 10)) TSA = 2(80 + 48 + 60) TSA = 2(188) TSA = 376 Step 5: State the final answer with units. Final Answer: The total surface area of the cuboidal box is 376 cm².
  • Example 2: Cylinder's Curved and Total Surface Area Find the curved surface area and the total surface area of a cylinder with radius 7 cm and height 10 cm. (Use π = 22/7) Solution: Step 1: Identify the shape as a cylinder. We need to find both CSA and TSA. Step 2: Given dimensions: r = 7 cm, h = 10 cm. Step 3: Formulas: CSA of cylinder = 2πrh TSA of cylinder = 2πr(h + r) Step 4: Substitute values and calculate: For CSA: CSA = 2 × (22/7) × 7 × 10 CSA = 2 × 22 × 10 CSA = 440 For TSA: TSA = 2 × (22/7) × 7 × (10 + 7) TSA = 2 × 22 × 17 TSA = 44 × 17 TSA = 748 Step 5: State the final answer with units. Final Answer: The curved surface area is 440 cm² and the total surface area is 748 cm².

Exam Tip: Avoiding Common Mistakes

When solving mensuration problems, small errors can lead to incorrect answers. Here are some common pitfalls to avoid:

  1. Confusing TSA with LSA/CSA: Always read the question carefully! If a problem asks for the 'area of four walls' or 'area to be painted excluding the floor and ceiling', you need LSA/CSA. If it says 'total surface area' or 'entire surface to be covered', use TSA.
  2. Incorrect Units: Ensure all given dimensions are in the same units before calculating. If not, convert them. Remember, surface area is always in square units (cm², m²).
  3. Calculation Errors with π: When working with cylinders, use the correct value of π (usually 22/7 or 3.14). Be careful with fractions and decimals in your calculations.
  4. Misinterpreting 'Open' Shapes: If a problem states a tank or box is 'open from the top', it means you should exclude the area of one base from the total surface area calculation. For a cylinder, this would be CSA + area of one circle (2πrh + πr²).
  5. Formula Recall: Practice writing down the formulas regularly. A slight mistake in a formula can derail your entire solution.

Practice Questions with Solutions

  • Q: A plastic box 1.5 m long, 1.25 m wide and 0.65 m deep is to be made. It is open at the top. Ignoring the thickness of the plastic sheet, determine the area of the sheet required for making the box. A: Step 1: Identify the shape as an open cuboid. We need to find the surface area of the box excluding the top face. Step 2: Given dimensions: l = 1.5 m, b = 1.25 m, h = 0.65 m. Step 3: The area of the sheet required will be the LSA of the cuboid plus the area of its base (bottom). Area = 2(l + b)h + lb Step 4: Substitute the values: 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 Final Answer: The area of the sheet required is 5.45 m².
  • 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. A: Step 1: Identify the shape as a rectangular hall (cuboid). We are given the perimeter of the floor and the cost of painting the four walls. Step 2: Given: Perimeter of floor = 2(l + b) = 250 m. Cost of painting = ₹15000. Rate = ₹10/m². Step 3: The area of four walls is the Lateral Surface Area (LSA) of the cuboid. We know that Cost = Area × Rate. So, LSA = Cost / Rate Also, LSA of cuboid = 2(l + b)h Step 4: Calculate LSA: LSA = 15000 / 10 = 1500 m² Now, substitute LSA and perimeter into the formula: 1500 = (2(l + b)) × h 1500 = 250 × h h = 1500 / 250 h = 6 Final Answer: The height of the hall is 6 m.
  • Q: The paint in a certain container is sufficient to paint an area equal to 9.375 m². How many bricks of dimensions 22.5 cm × 10 cm × 7.5 cm can be painted out of this container? A: Step 1: Identify the shape of a brick as a cuboid. The total paint available is given in m², so we need to find the total surface area of one brick in m². Step 2: Given dimensions of one brick: l = 22.5 cm = 0.225 m, b = 10 cm = 0.10 m, h = 7.5 cm = 0.075 m. Total paint area = 9.375 m². Step 3: Calculate the Total Surface Area (TSA) of one brick. TSA = 2(lb + bh + hl) Step 4: Substitute values for one brick: TSA = 2((0.225 × 0.10) + (0.10 × 0.075) + (0.075 × 0.225)) TSA = 2(0.0225 + 0.0075 + 0.016875) TSA = 2(0.046875) TSA = 0.09375 m² Number of bricks = Total paint area / TSA of one brick Number of bricks = 9.375 / 0.09375 Number of bricks = 100 Final Answer: 100 bricks can be painted out of the container.
  • Q: A closed cylindrical tank of radius 7 m and height 3 m is made from a sheet of metal. How much sheet of metal is required? A: Step 1: Identify the shape as a closed cylindrical tank. We need to find the Total Surface Area (TSA) of the cylinder. Step 2: Given dimensions: r = 7 m, h = 3 m. (Use π = 22/7). Step 3: The formula for TSA of a closed cylinder is 2πr(h + r). Step 4: Substitute the values: TSA = 2 × (22/7) × 7 × (3 + 7) TSA = 2 × 22 × 10 TSA = 44 × 10 TSA = 440 Final Answer: 440 m² of sheet metal is required.

Frequently Asked Questions

What is the key difference between Total Surface Area (TSA) and Lateral/Curved Surface Area (LSA/CSA)?

TSA includes the area of all faces/surfaces of a 3D object, including top and bottom. LSA/CSA only considers the area of the side faces, excluding the top and bottom, which is useful for 'area of four walls' or the curved part of a cylinder.

Why is surface area measured in square units?

Surface area is a measure of a two-dimensional extent, like the area of a flat shape. Since it's the sum of areas (length × breadth), its unit is always the square of the unit of length, such as cm² or m².

When should I use π = 22/7 versus π = 3.14?

Typically, you should use π = 22/7 when the radius or diameter is a multiple of 7, as it simplifies calculations. Otherwise, π = 3.14 is often used, especially if specified in the problem. Both are approximations.

Can surface area ever be negative?

No, surface area represents a physical measurement of space covered by the exterior of an object. As such, it must always be a non-negative value. If your calculation results in a negative number, it indicates a mistake in your steps or formula application.