Find The Width Of A Rectangle With Perimeter Of 64cm And Length Of 18 Cm is a common problem in geometry that involves understanding the relationship between the properties of a rectangle—specifically, its length, width, and perimeter. When given specific measurements, such as the perimeter and length, the goal is to determine the unknown dimension, which in this case is the width. This type of problem enhances our ability to apply formulas and develop problem-solving skills in real-world contexts, such as designing objects or understanding spatial dimensions.
In this article, we will explore step-by-step how to find the width of a rectangle when its perimeter and length are known. We will also delve into the formulas involved, provide practical examples, and discuss related concepts to strengthen your understanding of rectangle properties.
Understanding the Basic Properties of a Rectangle
Before diving into calculations, it is essential to understand the fundamental properties of rectangles:
What Is a Rectangle?
A rectangle is a four-sided polygon (quadrilateral) with opposite sides equal and all interior angles equal to 90 degrees. The key characteristics include:- Opposite sides are equal in length.
- The sum of all interior angles is 360 degrees.
- The diagonals are equal in length and bisect each other.
Important Measurements in a Rectangle
The primary measurements include:- Length (often denoted as L), which is the longer side.
- Width (or breadth, denoted as W), the shorter side.
- Perimeter (denoted as P), the total distance around the rectangle, calculated as the sum of all sides.
Formula for the Perimeter of a Rectangle
The perimeter of a rectangle is straightforward to compute with the formula:
P = 2 × (L + W)
where:
- P is the perimeter,
- L is the length,
- W is the width.
Given the perimeter and length, we can rearrange this formula to solve for the width.
Step-by-Step Calculation of the Width
Let's now apply this to the specific problem: a rectangle with a perimeter of 64 cm and a length of 18 cm.
Step 1: Write Down Known Values
- Perimeter, P = 64 cm
- Length, L = 18 cm
Step 2: Use the Perimeter Formula
Rearranged to solve for W:W = (P / 2) - L
Step 3: Substitute Known Values
Calculate:- P / 2 = 64 / 2 = 32
- Therefore,
Step 4: Final Answer
The width of the rectangle is 14 centimeters.Verification and Additional Considerations
It's always good practice to verify your solution:
- Calculate the perimeter using the found width:
- P = 2 × (L + W) = 2 × (18 + 14) = 2 × 32 = 64 cm
- Since the calculated perimeter matches the given perimeter, the solution is correct.
Furthermore, consider whether the dimensions make sense physically. Since both length and width are positive and less than the total perimeter divided by 2, the dimensions are reasonable.
Related Problems and Applications
Understanding how to find the width of a rectangle given perimeter and length is useful in many practical scenarios:
Design and Construction
- When designing a rectangular garden or room, knowing how to determine missing dimensions helps in planning materials and space utilization.
Manufacturing
- In manufacturing objects with specified perimeters, such as frames or panels, calculating dimensions accurately ensures proper fitting and aesthetics.
Mathematics Education
- These problems reinforce fundamental algebra skills and understanding of geometric formulas.
Additional Example: Finding Width When Perimeter and Length Vary
Suppose you are given a rectangle with:
- Perimeter, P = 50 cm
- Length, L = 12 cm
To find the width:
- Use the formula: W = (P / 2) - L
- Substitute: W = (50 / 2) - 12 = 25 - 12 = 13 cm
Thus, the width is 13 cm, and verifying:
- P = 2 × (12 + 13) = 2 × 25 = 50 cm, confirming the correctness.
Key Takeaways
- The perimeter of a rectangle is calculated as P = 2 × (L + W).
- Knowing the perimeter and length allows you to find the width using the formula W = (P / 2) - L.
- Always verify your calculations by plugging the value back into the perimeter formula.
- The concepts are applicable in real-world contexts such as design, construction, and manufacturing.
Conclusion
Finding the width of a rectangle given its perimeter and length involves straightforward algebra and understanding of basic geometric formulas. In the specific case of a rectangle with a perimeter of 64 cm and a length of 18 cm, the width is 14 cm. Mastering these calculations enhances problem-solving skills and provides a solid foundation for more advanced geometry topics. Whether you're designing a space, creating a craft project, or solving academic problems, knowing how to determine missing dimensions is a valuable skill that combines mathematical understanding with practical application.
