Calculating the Dimensions of a Rectangle Using Its Perimeter
Understanding the properties of a rectangle and how to calculate its dimensions using the perimeter is crucial for various applications. In this article, we will break down the process of finding the side lengths of a rectangle given its perimeter and one side length. Concepts such as perimeter, area, and the relationship between side lengths will be explained with clear examples and step-by-step solutions.
Introduction to Perimeter and Rectangle Dimensions
A rectangle is a four-sided polygon with opposite sides of equal length. The perimeter of a rectangle is the total length of its boundaries. The formula for the perimeter of a rectangle is:
P 2(l w)
where:
P is the perimeter l is the length w is the widthSolving for the Unknown Side of a Rectangle
Let's go through a specific example to illustrate how to solve for the unknown side of a rectangle when its perimeter and one side length are given. Our problem is as follows:
A rectangle has a perimeter of 30 inches. One side measures 5 inches. What are the measurements of the other side?
Step-by-Step Solution
Given: Perimeter P 30 inches and one side length l 5 inches. We need to find the width w.
Using the perimeter formula: P 2(l w), substitute the known values into the equation:
30 2(5 w)
Divide both sides of the equation by 2 to isolate the term with the unknown side length:
15 5 w
Subtract 5 from both sides to solve for w (the width of the rectangle):
10 w
Therefore, the length of the rectangle is 10 inches and the width is 5 inches.
Verifying the Solution
To ensure the solution is correct, we can substitute the values back into the perimeter formula:
P 2(l w) 2(10 5) 2(15) 30 inches
This confirms that the perimeter is indeed 30 inches, verifying our calculations.
Alternative Methods of Problem Solving
There are other ways to approach the problem as well. Here are a couple of examples:
Method 1: Examining Opposite Sides
In a rectangle, opposite sides are congruent. If one side is 5 inches, the opposite side will also be 5 inches. Therefore, the total length of these two sides is 5 5 10 inches. The remaining perimeter is:
30 - 10 20 inches
Since the remaining length is distributed between the other two sides, we can divide this by 2 to find the length of each of the remaining sides:
20 / 2 10 inches
Thus, the rectangle has two sides of 5 inches and two sides of 10 inches, confirming our solution.
Method 2: Using the Same Formula
Let Y be the unknown width. The formula for the perimeter becomes:
30 2(5 Y)
Follow the same steps as previously described to solve for Y:
The equation becomes:
30 2(5 Y)
Divide both sides by 2:
15 5 Y
Subtract 5 from both sides:
Y 10 inches
Thus, the other side length is 10 inches.
Conclusion
Understanding the relationship between the perimeter, length, and width of a rectangle is essential for various applications. By applying the formula and solving the equation step-by-step, we can easily find the dimensions of a rectangle. Whether using opposite sides or the perimeter formula, the key is to systematically isolate and solve for the unknown side.