Question

Express each statement as an equation involving the indicated variables. Perimeter of a Rectangle The perimeter \(P\) of a rectangle is twice the sum of its length \(l\) and its width \(w\).

Short answer

\( P = 2 \times (l + w) \)

Step-by-step solution

Identify Variables

The given problem mentions three variables: perimeter (P), length (l), and width (w).

Understand Perimeter Formula

The perimeter of a rectangle is defined as the total distance around the rectangle. For a rectangle, the perimeter is given by the sum of all its sides.

Formulate the Given Statement

According to the problem, the perimeter (P) of the rectangle is twice the sum of its length (l) and its width (w).

Express the Sum of Length and Width

The sum of the length and the width is represented as \( l + w \).

Multiply the Sum by Two

Since the perimeter is twice this sum, the expression becomes \( 2 \times (l + w) \).

Write the Equation

Equate the expression to the perimeter P: \( P = 2 \times (l + w) \).

Key concepts

Equation Formulation

To understand how to form an equation, let's break down the problem into a simpler narrative. Essentially, we need to translate a word problem into a mathematical equation. The statement we have is about the perimeter of a rectangle and how it relates to its length and width. We start by identifying the key parts of the statement and use logical steps to form the equation. Always remember, the goal is to represent the relationship using mathematical symbols and rules.

Variables

In any mathematical expression or equation, variables play a crucial role. They represent unknown or changeable values. In our rectangle problem, we have three key variables:

• Perimeter (P): This represents the total distance around the rectangle.
• Length (l): This is one of the longer sides of the rectangle.
• Width (w): This is one of the shorter sides of the rectangle.

By identifying these variables, we can easily input them into an equation to solve for one when the others are known. This step is foundational in setting up and solving geometry problems.

Perimeter Formula

The perimeter formula for a rectangle helps in calculating the total boundary length. For a rectangle, the perimeter is the sum of all its sides. A rectangle has two lengths and two widths. Thus, the perimeter is:
\[ P = 2l + 2w \]
However, to make it simpler, we can factor out the 2:
\[ P = 2 \times (l + w) \]
This formula essentially tells us that by adding the length and width and then doubling it, we get the rectangle's perimeter. It is a convenient and quick way to encapsulate the relationships between the dimensions of the rectangle.

Rectangle Geometry

Understanding the basic geometry of a rectangle is key to solving problems related to it. A rectangle is a four-sided figure with opposite sides equal and all angles being right angles (90 degrees). It has:

• Two lengths (longer sides)
• Two widths (shorter sides)

These properties are what define a rectangle and allow us to create formulas like the perimeter formula. Knowing these properties, especially that opposite sides are equal and all angles are right angles, simplifies many geometric calculations.