Ncert Class 10 Ex 13.1

NCERT Class 10 Ex 13.1: A Deep Dive into Surface Areas and Volumes

This article provides a comprehensive guide to NCERT Class 10 Chapter 13, Exercise 13.1, focusing on surface areas and volumes of three-dimensional shapes. We'll break down each problem, explaining the concepts and providing step-by-step solutions. This detailed approach will help you not only solve the exercise but also build a strong foundation in understanding surface areas and volumes, crucial for future mathematical studies. Understanding these concepts is essential for various applications, from engineering and architecture to everyday problem-solving.

Introduction to Surface Areas and Volumes

Before we delve into the specific problems in Exercise 13.1, let's refresh our understanding of surface area and volume.

• Surface Area: The total area of all the faces of a three-dimensional object. Think of it as the total area you would need to paint if you were to paint the entire object.

• Volume: The amount of space occupied by a three-dimensional object. It's the measure of how much the object can hold.

Exercise 13.1 primarily focuses on calculating surface areas and volumes of:

• Cuboids: Three-dimensional shapes with six rectangular faces.
• Cubes: Special cuboids where all six faces are squares.
• Cylinders: Three-dimensional shapes with two circular bases and a curved surface.

Understanding the Formulas

Mastering the formulas is key to successfully solving the problems in Exercise 13.1. Here's a summary of the essential formulas:

Cuboid:

• Surface Area (TSA): 2(lb + bh + hl), where l = length, b = breadth, h = height.
• Lateral Surface Area (LSA): 2h(l + b)
• Volume: l × b × h

Cube:

• Surface Area (TSA): 6a², where a = side length.
• Lateral Surface Area (LSA): 4a²
• Volume: a³

Cylinder:

• Surface Area (TSA): 2πr(r + h), where r = radius, h = height.
• Lateral Surface Area (LSA): 2πrh
• Volume: πr²h

Detailed Solutions to NCERT Class 10 Ex 13.1

Now let's tackle the problems in Exercise 13.1 step-by-step. Remember to always clearly state the given information and the formula you are using.

(Note: Since the exact problems in Exercise 13.1 are not provided, I will illustrate the solution approach with example problems representing the types of questions you'll encounter.)

Example Problem 1: Cuboid

Problem: A cuboidal water tank is 6m long, 5m wide, and 4m high. Find its (a) lateral surface area, (b) total surface area, and (c) volume.

Solution:

(a) Lateral Surface Area (LSA):

• Given: l = 6m, b = 5m, h = 4m
• Formula: LSA = 2h(l + b)
• Calculation: LSA = 2 × 4(6 + 5) = 8 × 11 = 88 m²

(b) Total Surface Area (TSA):

• Given: l = 6m, b = 5m, h = 4m
• Formula: TSA = 2(lb + bh + hl)
• Calculation: TSA = 2(6 × 5 + 5 × 4 + 4 × 6) = 2(30 + 20 + 24) = 2(74) = 148 m²

(c) Volume:

• Given: l = 6m, b = 5m, h = 4m
• Formula: Volume = l × b × h
• Calculation: Volume = 6 × 5 × 4 = 120 m³

Therefore, the lateral surface area is 88 m², the total surface area is 148 m², and the volume is 120 m³.

Example Problem 2: Cube

Problem: A cube has a side length of 7 cm. Calculate its surface area and volume.

Solution:

• Surface Area:

  • Given: a = 7 cm
  • Formula: TSA = 6a²
  • Calculation: TSA = 6 × 7² = 6 × 49 = 294 cm²

• Volume:

  • Given: a = 7 cm
  • Formula: Volume = a³
  • Calculation: Volume = 7³ = 343 cm³

Therefore, the surface area of the cube is 294 cm², and its volume is 343 cm³.

Example Problem 3: Cylinder

Problem: A cylindrical water pipe has a radius of 5 cm and a height of 100 cm. Find its (a) curved surface area, (b) total surface area, and (c) volume.

Solution:

(a) Curved Surface Area (CSA):

• Given: r = 5 cm, h = 100 cm
• Formula: CSA = 2πrh
• Calculation: CSA = 2 × π × 5 × 100 = 1000π cm² (You can use the value of π as 22/7 or 3.14)

(b) Total Surface Area (TSA):

• Given: r = 5 cm, h = 100 cm
• Formula: TSA = 2πr(r + h)
• Calculation: TSA = 2 × π × 5(5 + 100) = 10π × 105 = 1050π cm²

(c) Volume:

• Given: r = 5 cm, h = 100 cm
• Formula: Volume = πr²h
• Calculation: Volume = π × 5² × 100 = 2500π cm³

Common Mistakes and How to Avoid Them

• Incorrect Formula Selection: Double-check that you are using the correct formula for the shape and the property (surface area or volume) you need to calculate.

• Unit Confusion: Ensure consistent units throughout your calculations. If the dimensions are given in centimeters, your answer should be in cubic centimeters for volume and square centimeters for surface area.

• Calculation Errors: Carefully perform each step of your calculations. Use a calculator if needed, but always double-check your work.

• Forgetting π: Remember to include π (pi) in your calculations when dealing with cylinders.

Frequently Asked Questions (FAQ)

Q1: What is the difference between lateral surface area and total surface area?

A1: Lateral surface area is the area of the sides of a three-dimensional object, excluding the top and bottom faces. Total surface area includes the area of all faces, including the top and bottom.

Q2: Why is understanding surface area and volume important?

A2: Understanding surface area and volume is essential in various fields like architecture (calculating material needed for construction), engineering (designing efficient structures), and even everyday situations (determining the capacity of containers).

Q3: Can I use a calculator for this exercise?

A3: Yes, using a calculator is allowed and recommended, especially for complex calculations involving π. However, it's crucial to show your work clearly, indicating the formula used and the steps involved.

Q4: What if the problem involves a combination of shapes?

A4: Problems involving combined shapes require you to break them down into individual shapes (cuboids, cylinders, etc.), calculate the surface area and volume of each individual shape, and then add or subtract accordingly, depending on the problem's context.

Conclusion

Mastering Exercise 13.1 requires a solid understanding of the formulas for surface area and volume of cuboids, cubes, and cylinders. By following the step-by-step approach outlined in this article, practicing with various problems, and carefully reviewing the common mistakes to avoid, you'll build a strong foundation in these crucial geometrical concepts. Remember to always clearly define the given information, choose the appropriate formula, and carefully perform the calculations. With practice and attention to detail, you'll confidently solve any problem related to surface areas and volumes. Good luck!