How to find a simplified expression for the perimeter of a rectangle?

To find a simplified expression for the perimeter of a rectangle, you combine the lengths of all its sides, resulting in the common formula $2L + 2W$ or $2(L + W)$, where L represents the length and W represents the width.

Understanding Rectangle Perimeter Basics

The perimeter of any two-dimensional shape is the total distance around its outer boundary. For a rectangle, this involves adding the lengths of its four sides. Rectangles are characterized by having two pairs of equal sides: two lengths and two widths.

Imagine a rectangle with specific dimensions. If one side is denoted as x (representing length) and the adjacent side as y (representing width), the four sides of the rectangle would be x, y, x, and y.

Step-by-Step Simplification

Simplifying the expression for a rectangle's perimeter involves combining like terms, which makes the formula more concise and practical for calculations.

1. Identify the Sides

A rectangle has two lengths and two widths.

• Let Length be represented by the variable L.
• Let Width be represented by the variable W.

2. Write the Initial Expression

The perimeter ($P$) is the sum of all its sides. If you trace around the rectangle, you would add one length, then one width, then another length, and finally another width.
$P = L + W + L + W$

3. Combine Like Terms

To simplify, group the similar terms together (the lengths with lengths, and the widths with widths).
$P = (L + L) + (W + W)$
Adding the like terms:
$P = 2L + 2W$

This step directly applies the principle that if you have x + y + x + y, combining those terms yields 2x + 2y.

4. Factor (Optional but Common Simplification)

An additional way to simplify the expression is to factor out the common number, which in this case is 2.
$P = 2(L + W)$

This factored form clearly shows that you can find the perimeter by adding the length and width first, and then doubling that sum.

Why Simplify?

Simplifying the perimeter expression offers several advantages, making calculations more efficient and the formula easier to understand and apply.

• Efficiency: Simplified formulas require fewer steps, leading to quicker calculations.
• Clarity: The condensed form (2L + 2W or 2(L + W)) intuitively shows that you need two lengths and two widths to form the perimeter.
• Generalization: A simplified formula provides a universal rule that applies to any rectangle, regardless of its specific dimensions, making it a foundational concept in geometry.

Practical Examples

Let's illustrate how to use the simplified expressions with some examples.

Example 1: Using Numerical Values

Consider a rectangle with a length of 8 units and a width of 5 units.

• Using the initial sum:
  $P = 8 + 5 + 8 + 5 = 26$ units
• Using the simplified form ($2L + 2W$):
  $P = 2(8) + 2(5)$
  $P = 16 + 10 = 26$ units
• Using the factored form ($2(L + W)$):
  $P = 2(8 + 5)$
  $P = 2(13) = 26$ units

As seen, all methods yield the same correct perimeter, but the simplified formulas are more direct.

Example 2: With Variable Expressions

Suppose a rectangle has a length of $(3x + 1)$ and a width of $(x - 2)$.

To find the simplified expression for its perimeter:
$P = 2L + 2W$
Substitute the given expressions for L and W:
$P = 2(3x + 1) + 2(x - 2)$
Distribute the 2 to each term inside the parentheses:
$P = (2 \cdot 3x) + (2 \cdot 1) + (2 \cdot x) + (2 \cdot -2)$
$P = 6x + 2 + 2x - 4$
Combine like terms (x terms together and constant terms together):
$P = (6x + 2x) + (2 - 4)$
$P = 8x - 2$

This is the simplified expression for the perimeter of the rectangle with those variable dimensions.

For more details on simplifying expressions in general, you can explore resources like Khan Academy's lessons on combining like terms.