The perimeter is \(2(w + 2w) = 72\).

["Understanding the Perimeter: Solving (2(w + 2w) = 72) Step-by-Step", "When solving geometric formulas, understanding the perimeter of a rectangle is both fundamental and practical. In this article, we break down the key problem: (2(w + 2w) = 72), exploring how to find the value of (w) and use it to calculate the full perimeter. This straightforward equation reveals essential algebraic thinking essential for geometry, algebra, and real-world applications.", "---", "### What Is the Perimeter of a Rectangle?", "The perimeter refers to the total distance around the outside of a shape. For a rectangle, the perimeter (P) is given by:\n[\nP = 2 \ imes (\ ext{length} + \ ext{width})\n]\nThis formula is crucial for determining the boundary length of fences, borders, flooring, or any enclosed rectangular area.", "---", "### Step-by-Step: Solving (2(w + 2w) = 72)", "Let’s solve the equation step-by-step to uncover the dimensions and confirm the perimeter.", "1. Combine Like Terms Inside the Parentheses\n The expression (w + 2w) simplifies to (3w):\n [\n 2(w + 2w) = 2(3w) = 6w\n ]\n So the equation becomes:\n [\n 6w = 72\n ]", "2. Solve for (w)\n Divide both sides by 6 to isolate (w):\n [\n w = \frac{72}{6} = 12\n ]", "3. Determine Length and Width\n Since (w) represents the width, and length is given as (2w):\n [\n \ ext{width} = w = 12\n ]\n [\n \ ext{length} = 2w = 2 \ imes 12 = 24\n ]", "4. Calculate the Perimeter\n Plug into the perimeter formula:\n [\n P = 2(\ ext{length} + \ ext{width}) = 2(24 + 12) = 2 \ imes 36 = 72\n ]\n This confirms the original equation:\n [\n 2(w + 2w) = 72\n ]", "---", "### Why This Equation Matters", "Solving (2(w + 2w) = 72) not only confirms the perimeter is 72 units but also demonstrates how algebra supports geometry. This type of problem is common in:", "- Classroom geometry exercises\n- Architecture and construction planning\n- Design and landscaping projects\n- Everyday scenarios involving fencing or room dimensions", "Understanding how to translate a geometric layout into an algebraic expression – and solve it – builds critical thinking skills valuable across STEM fields.", "---", "### Final Answer", "The perimeter of the rectangle described by (2(w + 2w) = 72) is:\n[\n\boxed{72}\n]\nThe width is (12), and the length is (24), confirming the perimeter using algebra and geometry fundamentals.", "---", "### Pro Tips for Mastering Perimeters", "- Always simplify expressions inside parentheses before expanding.\n- Recognize the relationship between dimensions (e.g., length = (2w)) to avoid confusion.\n- Practice converting word problems into equations—it enhances problem-solving fluency.\n- Use real-life examples (like measuring a garden or room) to reinforce understanding.", "---", "Mastering the perimeter equation (2(w + 2w) = 72) is a gateway to solving more complex geometric problems. With practice, algebra becomes a powerful tool for understanding spatial relationships in everyday life."]