The Perimeter Of A Rectangle Is 20.4 Dm, If One Of Its Sides Measures 6.3 Dm, Find The Area.

In geometry, a rectangle is a four-sided polygon with four right angles. Its opposite sides are equal in length. The perimeter of a rectangle is the total distance around its outside, which is the sum of the lengths of all four sides. The area of a rectangle, on the other hand, is the amount of surface it covers, and it's calculated by multiplying its length and width. Understanding how to calculate the perimeter and area of rectangles is crucial in various fields, including architecture, engineering, and everyday problem-solving.

This article addresses a common problem encountered in geometry: determining the area of a rectangle when its perimeter and the length of one side are known. This type of problem often appears in national exams and tests understanding of fundamental geometric principles and algebraic manipulation. The ability to solve such problems is not only essential for academic success but also for practical applications in real-world scenarios.

The problem we will explore involves a rectangle with a given perimeter of 20.4 decimeters (dm) and one side measuring 6.3 dm. Our goal is to find the area of this rectangle. To achieve this, we will first use the given information to calculate the length of the other side. Once we have both the length and the width, we can easily determine the area using the formula for the area of a rectangle. This process involves a combination of geometric understanding and algebraic problem-solving skills.

Before diving into the solution, it's crucial to have a solid understanding of the concepts of perimeter and area. The perimeter is the total distance around the outside of a two-dimensional shape. For a rectangle, this means adding up the lengths of all four sides. Since a rectangle has two pairs of equal sides (length and width), the formula for the perimeter (P) can be expressed as:

P = 2l + 2w

Where:

• l represents the length of the rectangle.
• w represents the width of the rectangle.

The area, on the other hand, measures the amount of surface a shape covers. For a rectangle, the area (A) is calculated by multiplying its length and width. The formula for the area of a rectangle is:

A = l * w

Where:

• l represents the length of the rectangle.
• w represents the width of the rectangle.

These two concepts, perimeter and area, are fundamental in geometry and have numerous practical applications. For example, if you were fencing a rectangular garden, you would need to calculate the perimeter to determine how much fencing material to purchase. If you were laying sod in the same garden, you would need to calculate the area to determine how much sod to buy. The ability to differentiate between perimeter and area and to apply the correct formulas is essential for solving geometric problems accurately.

In the context of our problem, we are given the perimeter and one side of the rectangle. To find the area, we must first use the perimeter formula to determine the length of the missing side. This involves algebraic manipulation and a clear understanding of the relationships between the sides and the perimeter of a rectangle. Once we have both sides, calculating the area is a straightforward application of the area formula.

In this specific problem, we are given that the perimeter of the rectangle is 20.4 dm and one of its sides measures 6.3 dm. Let's assume that the given side is the width (w) of the rectangle. Our first task is to find the length (l). We can use the perimeter formula to solve for the missing side:

P = 2l + 2w

We know that P = 20.4 dm and w = 6.3 dm. Substituting these values into the formula, we get:

20.4 = 2l + 2(6.3)

Now, we need to solve this equation for l. First, we can simplify the equation by multiplying 2 by 6.3:

20.4 = 2l + 12.6

Next, we subtract 12.6 from both sides of the equation to isolate the term with l:

20.4 - 12.6 = 2l

7.  8 = 2l

Finally, we divide both sides of the equation by 2 to solve for l:

7.  8 / 2 = l

l = 3.9 dm

So, the length of the rectangle is 3.9 dm. Now that we have both the length and the width, we can proceed to calculate the area of the rectangle. This step demonstrates the importance of algebraic skills in solving geometric problems. By understanding and applying the perimeter formula, we were able to determine the missing side, which is a crucial step in finding the area.

Now that we have determined both the length (l = 3.9 dm) and the width (w = 6.3 dm) of the rectangle, we can easily calculate the area. The formula for the area of a rectangle is:

A = l * w

Substituting the values we found for l and w into the formula, we get:

A = 3.9 dm * 6.3 dm

Multiplying these two values together, we find:

A = 24.57 square dm

Therefore, the area of the rectangle is 24.57 square decimeters (dm²). This result provides the final answer to the problem. The process of calculating the area involves a straightforward application of the area formula, but it relies on the accurate determination of the length and width. In this case, we first had to use the perimeter formula to find the missing side before we could calculate the area.

The unit for area is square decimeters (dm²) because we are measuring the amount of surface covered by the rectangle. It's essential to include the correct units in your answer to ensure clarity and accuracy. This example highlights the importance of understanding both the formulas for geometric shapes and the units in which measurements are expressed.

In summary, we have successfully calculated the area of a rectangle given its perimeter and one side length. The problem required us to utilize the formulas for both the perimeter and the area of a rectangle. First, we used the perimeter formula to solve for the missing side, and then we used the area formula to find the area. The final answer, as we determined, is 24.57 square decimeters (dm²).

This exercise demonstrates a fundamental approach to solving geometric problems. It involves:

1. Understanding the relevant formulas and concepts.
2. Using given information to set up equations.
3. Solving for unknown variables.
4. Applying the results to find the desired quantity.

These steps are not only applicable to this specific problem but also to a wide range of geometric and mathematical challenges. Mastering these skills is essential for success in mathematics and related fields.

The ability to solve problems like this is important for various reasons. Firstly, it reinforces your understanding of geometric principles and formulas. Secondly, it enhances your algebraic problem-solving skills. Finally, it prepares you for more complex problems in geometry and other areas of mathematics.

Moreover, the concepts of perimeter and area have practical applications in everyday life. Whether you are calculating the amount of paint needed for a wall, the amount of fencing required for a garden, or the amount of flooring needed for a room, understanding these concepts is crucial. Therefore, mastering these skills is not only beneficial for academic purposes but also for real-world problem-solving.

The problem-solving approach we used in this article can be applied to a variety of geometric problems. By carefully analyzing the given information, identifying the relevant formulas, and systematically solving for unknowns, you can successfully tackle a wide range of challenges. Practice and familiarity with these techniques will build your confidence and improve your problem-solving abilities in geometry and beyond.