Its Breadth. Which Q7: A Wire In The Shape Of A Rectangle. Its Length Is 40 Cm And Breadth Is 22 Cm. If The Same Wire Is Rebent In The Shape Of A Square, What Will Be The Measure Of Each Side? Also , Find Which Shape Encloses More Area? Q8: A Wire Of

Mar 8, 2025 by ADMIN 251 views

**Mathematical Problem Solving: Understanding the Relationship Between Rectangles and Squares**

Introduction

In mathematics, we often come across problems that involve shapes and their properties. One such problem is the comparison of the areas of a rectangle and a square when the same wire is used to form both shapes. In this article, we will explore this problem and provide a step-by-step solution to find the measure of each side of the square and determine which shape encloses more area.

Problem Statement

A wire in the shape of a rectangle has a length of 40 cm and a breadth of 22 cm. If the same wire is rebent in the shape of a square, what will be the measure of each side? Also, find which shape encloses more area.

Solution

To solve this problem, we need to first find the perimeter of the rectangle. The perimeter of a rectangle is given by the formula:

Perimeter = 2(length + breadth)

Substituting the given values, we get:

Perimeter = 2(40 + 22) Perimeter = 2(62) Perimeter = 124 cm

Since the same wire is used to form the square, the perimeter of the square will also be 124 cm. The perimeter of a square is given by the formula:

Perimeter = 4(side)

Equating the perimeter of the square to 124 cm, we get:

4(side) = 124 side = 124/4 side = 31 cm

Therefore, the measure of each side of the square is 31 cm.

Comparing the Areas of the Rectangle and the Square

Now that we have found the measure of each side of the square, we can compare the areas of the rectangle and the square. The area of a rectangle is given by the formula:

Area = length × breadth

Substituting the given values, we get:

Area = 40 × 22 Area = 880 cm²

The area of a square is given by the formula:

Area = side²

Substituting the value of the side, we get:

Area = 31² Area = 961 cm²

Comparing the areas of the rectangle and the square, we can see that the square encloses more area than the rectangle.

Conclusion

In this article, we solved a problem that involved finding the measure of each side of a square when the same wire is used to form both a rectangle and a square. We also compared the areas of the rectangle and the square and found that the square encloses more area than the rectangle. This problem is a great example of how mathematical concepts can be applied to real-world scenarios.

Mathematical Concepts Used

Perimeter of a rectangle
Perimeter of a square
Area of a rectangle
Area of a square

Real-World Applications

This problem has real-world applications in various fields such as engineering, architecture, and design. For example, in engineering, the design of a structure such as a bridge or a building requires the calculation of the perimeter and area of the structure. In architecture, the design of a building requires the calculation of the perimeter and area of the building. In design, the calculation of the perimeter and area of a product is essential in determining its size and shape.

Tips and Tricks

When solving problems involving shapes, it is essential to use the correct formulas and units.
When comparing the areas of different shapes, it is essential to use the same units.
When solving problems involving real-world applications, it is essential to consider the constraints and limitations of the problem.

Frequently Asked Questions

Q: What is the perimeter of a rectangle? A: The perimeter of a rectangle is given by the formula: Perimeter = 2(length + breadth)
Q: What is the perimeter of a square? A: The perimeter of a square is given by the formula: Perimeter = 4(side)
Q: What is the area of a rectangle? A: The area of a rectangle is given by the formula: Area = length × breadth
Q: What is the area of a square? A: The area of a square is given by the formula: Area = side²
Mathematical Problem Solving: Understanding the Relationship Between Rectangles and Squares ====================================================================================

Q&A: Frequently Asked Questions

Q1: What is the difference between the perimeter and area of a rectangle and a square?

A1: The perimeter of a rectangle and a square is the distance around the shape, while the area is the amount of space inside the shape. The perimeter of a rectangle is given by the formula: Perimeter = 2(length + breadth), while the area is given by the formula: Area = length × breadth. The perimeter of a square is given by the formula: Perimeter = 4(side), while the area is given by the formula: Area = side².

Q2: How do you find the measure of each side of a square when the same wire is used to form both a rectangle and a square?

A2: To find the measure of each side of a square when the same wire is used to form both a rectangle and a square, you need to first find the perimeter of the rectangle. The perimeter of a rectangle is given by the formula: Perimeter = 2(length + breadth). Since the same wire is used to form the square, the perimeter of the square will also be the same. The perimeter of a square is given by the formula: Perimeter = 4(side). Equating the perimeter of the square to the perimeter of the rectangle, you can find the measure of each side of the square.

Q3: What is the relationship between the perimeter and area of a rectangle and a square?

A3: The perimeter and area of a rectangle and a square are related in that the perimeter of a rectangle is equal to the perimeter of a square with the same side length. However, the area of a rectangle is not equal to the area of a square with the same side length. The area of a rectangle is given by the formula: Area = length × breadth, while the area of a square is given by the formula: Area = side².

Q4: How do you compare the areas of a rectangle and a square?

A4: To compare the areas of a rectangle and a square, you need to first find the area of each shape. The area of a rectangle is given by the formula: Area = length × breadth, while the area of a square is given by the formula: Area = side². Once you have found the areas of both shapes, you can compare them to determine which shape encloses more area.

Q5: What are some real-world applications of the concepts of perimeter and area?

A5: The concepts of perimeter and area have many real-world applications in various fields such as engineering, architecture, and design. For example, in engineering, the design of a structure such as a bridge or a building requires the calculation of the perimeter and area of the structure. In architecture, the design of a building requires the calculation of the perimeter and area of the building. In design, the calculation of the perimeter and area of a product is essential in determining its size and shape.

Q6: How do you use mathematical concepts to solve problems involving shapes?

A6: To use mathematical concepts to solve problems involving shapes, you need to first identify the type of problem and the relevant mathematical concepts. For example, if the problem involves finding the perimeter of a rectangle, you need to use the formula: Perimeter = 2(length + breadth). If the problem involves finding the area of a square, you need to use the formula: Area = side². Once you have identified the relevant mathematical concepts, you can apply them to solve the problem.

Q7: What are some tips and tricks for solving problems involving shapes?

A7: Some tips and tricks for solving problems involving shapes include:

Using the correct formulas and units
Checking your work for errors
Using visual aids such as diagrams and graphs to help you understand the problem
Breaking down complex problems into simpler sub-problems
Using mathematical concepts to check your work and ensure that your answer is correct.

Q8: How do you use mathematical concepts to compare the areas of different shapes?

A8: To use mathematical concepts to compare the areas of different shapes, you need to first find the area of each shape. The area of a rectangle is given by the formula: Area = length × breadth, while the area of a square is given by the formula: Area = side². Once you have found the areas of both shapes, you can compare them to determine which shape encloses more area.

Q9: What are some common mistakes to avoid when solving problems involving shapes?

A9: Some common mistakes to avoid when solving problems involving shapes include:

Using the wrong formulas or units
Failing to check your work for errors
Not using visual aids such as diagrams and graphs to help you understand the problem
Not breaking down complex problems into simpler sub-problems
Not using mathematical concepts to check your work and ensure that your answer is correct.

Q10: How do you use mathematical concepts to solve problems involving real-world applications?

A10: To use mathematical concepts to solve problems involving real-world applications, you need to first identify the type of problem and the relevant mathematical concepts. For example, if the problem involves designing a building, you need to use mathematical concepts such as perimeter and area to determine the size and shape of the building. Once you have identified the relevant mathematical concepts, you can apply them to solve the problem.