Finding The Area Of A Rectangle: A Step-by-Step Guide

Hey guys! Let's dive into a classic geometry problem and figure out how to find the area of a rectangle. We're given a specific scenario: a rectangle with a perimeter of 32 centimeters and a width of 7 centimeters. Our mission? To calculate the area and choose the correct answer from the options provided. It's a fun puzzle that's all about understanding the relationship between perimeter, width, length, and area. This is a common type of question you might find in math tests or simply when you're exploring the world of shapes. Understanding how to solve this is super important. We will break down the steps, making sure everything is clear and easy to follow. Get ready to flex those math muscles and discover how to find the area of our rectangle. The problem isn’t as hard as it might seem at first glance, so let's get started!

Understanding the Basics: Perimeter, Width, and Length

Before we start calculating the area, it's essential to understand the basics of what we're working with. Perimeter is the total distance around the outside of a shape. Imagine walking around the rectangle; the perimeter is the total distance you'd walk. For a rectangle, we calculate the perimeter by adding up the lengths of all four sides. Since opposite sides of a rectangle are equal, the formula for the perimeter is 2 * (length + width). We're given that the perimeter of our rectangle is 32 centimeters. This is our starting point. The width of a rectangle is the shorter side, and in our problem, we know the width is 7 centimeters. The length, on the other hand, is the longer side of the rectangle. This is what we need to find first. Once we know the length, finding the area will be a piece of cake. Knowing the width and the perimeter, we can use the perimeter formula to calculate the length. This will allow us to move on to finding the area. Remember, the area measures the space inside the rectangle. The relationship between perimeter, width, and length is fundamental to solving this problem. Think of it as a set of interconnected pieces, and once you figure out one, the others fall into place.

Calculating the Length: Unraveling the Puzzle

Now that we've got the basics down, let's calculate the length of the rectangle. We know the perimeter (P) is 32 cm, and the width (w) is 7 cm. The formula for the perimeter is P = 2 * (length + width), which we can also write as P = 2l + 2w. Let's substitute the known values into the formula: 32 = 2l + 2 * 7. This simplifies to 32 = 2l + 14. To find the length (l), we first subtract 14 from both sides of the equation: 32 - 14 = 2l. This gives us 18 = 2l. Now, we divide both sides by 2 to solve for l: 18 / 2 = l. Therefore, the length (l) is 9 cm. So we have now found the length of the rectangle. We’re one step closer to solving the problem. The length is a crucial value for calculating the area. Knowing both the length and width is all we need to find the final answer. This step-by-step approach ensures that you don't miss any part of the calculation. With the length calculated, we can confidently proceed to the next step.

Finding the Area: The Grand Finale

Alright, we've got the length and the width. Now, let's find the area. The area (A) of a rectangle is calculated using the formula: A = length * width (A = l * w). We know the length (l) is 9 cm and the width (w) is 7 cm. So, A = 9 cm * 7 cm. Multiplying these values gives us A = 63 cm². Now, compare our answer with the given options: A) 56 cm² B) 64 cm² C) 72 cm² D) 48 cm². None of the options match our calculated area of 63 cm². This may suggest a mistake, or we might need to double check our calculations. However, we've worked through the problem step by step, and the closest option is likely to be the one that is closest to our answer. This final step is straightforward, but it's important not to rush. Always double-check your calculations. The area represents the space the rectangle occupies, making it an essential concept in geometry. Keep in mind that when calculating the area, the units are squared (cm²), which represents the two-dimensional space. We must choose the option closest to our answer.

Reviewing the Options and Final Answer

We calculated the area of the rectangle to be 63 cm². Looking back at the options, we see that none of the answers match our calculation exactly. This could mean there might have been a slight error in the problem, or perhaps we need to reconsider our approach. However, given the options, let's review the closest ones. A) 56 cm² is not correct. B) 64 cm² is also not the correct answer, but it is closer. C) 72 cm² is not the correct answer. D) 48 cm² is incorrect. Since none of the answers are correct, it may be due to an error in the given options. However, as the closest option is B) 64 cm², as it is the closest value. Always remember to double-check your work and consider all possibilities. We have successfully worked through the problem, demonstrating how to find the length and subsequently the area of a rectangle. This process showcases the importance of understanding formulas and applying them step by step. We have successfully learned how to solve the problem and have gone through the steps and formula to find the solution. Even though the correct answer was not on the options, this demonstrates the importance of the correct methodology and how to approach the problem.

Conclusion: Mastering the Rectangle Area Calculation

So there you have it, guys! We've successfully navigated the process of finding the area of a rectangle when given its perimeter and width. We started with the basics, understanding the concepts of perimeter, width, and length. Then, we calculated the length using the perimeter formula. After that, we used the length and width to determine the area. Even though the correct answer was not provided in the options, the process is what matters. This problem is a great example of how mathematical concepts are interconnected and how understanding one part can lead to solving the whole. Remember, practice is key! The more you work through these types of problems, the easier they become. Don’t be afraid to break down the problem into smaller, manageable steps. By understanding the formulas and the relationship between the different elements of a rectangle, you can confidently solve any similar problem. Keep practicing, and you'll become a pro at these calculations in no time! Keep exploring and have fun with math!