Class 10 Maths Exercise 13.1

Mastering Class 10 Maths Exercise 13.1: Surface Areas and Volumes of Cubes and Cuboids

This complete walkthrough digs into Class 10 Maths Exercise 13.So 1, focusing on the surface areas and volumes of cubes and cuboids. Understanding these fundamental 3D shapes is crucial for building a strong foundation in geometry and preparing for higher-level mathematics. We'll cover the key formulas, provide step-by-step solutions to example problems, and address common student queries to ensure you master this chapter with confidence. This article aims to be your one-stop resource for conquering Exercise 13.1, making complex concepts clear and accessible.

Introduction to Cubes and Cuboids

Before diving into the problems, let's refresh our understanding of cubes and cuboids. Both are three-dimensional shapes, but they differ in their dimensions:

• Cube: A cube is a three-dimensional shape with six identical square faces. All its sides (length, width, and height) are equal in length.

• Cuboid: A cuboid is a three-dimensional shape with six rectangular faces. It has three pairs of opposite faces that are equal in size and shape. The length, width, and height can be different.

Key Formulas for Surface Area and Volume

To solve the problems in Exercise 13.1, we need to understand the following formulas:

For a Cube:

• Surface Area (SA): 6a², where 'a' is the length of one side.
• Volume (V): a³

For a Cuboid:

• Surface Area (SA): 2(lb + bh + hl), where 'l' is length, 'b' is breadth (width), and 'h' is height.
• Volume (V): lbh

Step-by-Step Solutions to Example Problems

Let's work through some example problems similar to those found in Exercise 13.1. These examples will illustrate the application of the formulas and help you understand the problem-solving process.

Example 1: Finding the Surface Area of a Cube

A cube has a side of 5 cm. Find its surface area.

Solution:

1. Identify the given: The side (a) of the cube is 5 cm.
2. Apply the formula: Surface Area (SA) = 6a²
3. Substitute the value: SA = 6 * (5 cm)² = 6 * 25 cm² = 150 cm²
4. State the answer: The surface area of the cube is 150 cm².

Example 2: Finding the Volume of a Cuboid

A cuboid has a length of 10 cm, a breadth of 6 cm, and a height of 4 cm. Find its volume.

Solution:

1. Identify the given: Length (l) = 10 cm, Breadth (b) = 6 cm, Height (h) = 4 cm.
2. Apply the formula: Volume (V) = lbh
3. Substitute the values: V = 10 cm * 6 cm * 4 cm = 240 cm³
4. State the answer: The volume of the cuboid is 240 cm³.

Example 3: A Combined Problem

A room is in the shape of a cuboid. Its length is 5 meters, breadth is 4 meters, and height is 3 meters. Find the cost of painting the four walls and the ceiling at a rate of ₹20 per square meter.

Solution:

1. Calculate the area of the four walls:

   • Area of four walls = 2h(l + b) = 2 * 3 m * (5 m + 4 m) = 54 m²

2. Calculate the area of the ceiling:

   • Area of ceiling = lb = 5 m * 4 m = 20 m²

3. Calculate the total area to be painted:

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   • Total area = Area of four walls + Area of ceiling = 54 m² + 20 m² = 74 m²

4. Calculate the cost of painting:

   • Cost = Total area * Rate per square meter = 74 m² * ₹20/m² = ₹1480

5. State the answer: The cost of painting the four walls and the ceiling is ₹1480.

Example 4: Problem Involving a Change in Dimensions

A cube has a side of 7cm. If the side is increased by 2cm, find the increase in its surface area and volume.

Solution:

1. Original Cube:

   • Original surface area = 6 * (7cm)² = 294 cm²
   • Original volume = (7cm)³ = 343 cm³

2. New Cube:

   • New side = 7cm + 2cm = 9cm
   • New surface area = 6 * (9cm)² = 486 cm²
   • New volume = (9cm)³ = 729 cm³

3. Increase in surface area:

   • Increase = 486 cm² - 294 cm² = 192 cm²

4. Increase in volume:

   • Increase = 729 cm³ - 343 cm³ = 386 cm³

5. State the answer: The increase in surface area is 192 cm², and the increase in volume is 386 cm³.

Explanation of Scientific Principles

The formulas for surface area and volume are derived from basic geometric principles. The surface area represents the total area of all the faces of the 3D shape. The volume represents the amount of space enclosed within the 3D shape. For a cube, all faces are identical squares, leading to the simple formula 6a². Plus, for a cuboid, the faces are rectangles, requiring a more detailed formula that accounts for the different dimensions. These formulas are fundamental building blocks for understanding more complex geometric calculations later in your studies.

Frequently Asked Questions (FAQ)

• Q: What is the difference between a cube and a cuboid?

  • A: A cube has all sides equal in length, while a cuboid has different lengths for at least two of its sides.

• Q: What are the units for surface area and volume?

  • A: Surface area is measured in square units (e.g., cm², m²), while volume is measured in cubic units (e.g., cm³, m³).

• Q: How do I handle problems involving units of measurement?

  • A: Always confirm that all measurements are in the same units before applying the formulas. If necessary, convert between units (e.g., centimeters to meters) before calculation.

• Q: What if the problem involves a shape that is not a perfect cube or cuboid?

  • A: More complex shapes often require breaking them down into simpler cuboid or cube-like shapes to calculate the surface area and volume.

Conclusion

Mastering Exercise 13.Don't hesitate to revisit these examples and practice additional problems to solidify your understanding. That's why remember to systematically approach each problem, carefully identifying the given information and applying the correct formula. Here's the thing — 1 requires a thorough understanding of the formulas for surface area and volume of cubes and cuboids. By practicing the problems and understanding the underlying geometric principles, you can confidently tackle more complex problems in three-dimensional geometry. With consistent practice and a clear understanding of the concepts, success in this chapter, and future mathematical endeavors, is well within your reach. Good luck!