The perimeter is \( 2(w + 2w) = 6w = 36 \).

["Understanding Perimeters: Solving ( 2(w + 2w) = 6w = 36 ) Step-by-Step", "When learning geometry, understanding how to calculate the perimeter of a shape based on its dimensions is fundamental. One common problem involves recognizing the perimeter formula for certain geometric figures, especially rectangles, and solving equations related to it. Today, we’ll break down the equation ( 2(w + 2w) = 6w = 36 ) step-by-step to clarify how to solve perimeter-related problems efficiently.", "### What Is Perimeter?", "Perimeter is the total distance around the boundary of a two-dimensional shape. For a rectangle, the perimeter formula is:", "[\n\ ext{Perimeter} = 2 \ imes (\ ext{length} + \ ext{width})\n]", "In this problem, we see the length expressed in terms of ( w ) as ( 2w ), and the width is simply ( w ). Thus, the full calculation follows naturally from applying the perimeter formula.", "### Step 1: Recognize the Shape and Use the Perimeter Formula", "Since one side of the rectangle is ( w ) and the adjacent side is ( 2w ), the figure’s perimeter is calculated using two copies of ( (w + 2w) ):", "[\n2(w + 2w)\n]", "Simplifying inside the parentheses:\n[\nw + 2w = 3w\n]", "Then multiply by 2:\n[\n2 \ imes 3w = 6w\n]", "This confirms the given perimeter expression is simplified as ( 6w ).", "### Step 2: Set the Expression Equal to the Given Perimeter", "We are told the perimeter is 36, so:\n[\n6w = 36\n]", "### Step 3: Solve for ( w )", "To find ( w ), divide both sides of the equation by 6:\n[\nw = \frac{36}{6} = 6\n]", "### Step 4: Verify the Solution", "Plug ( w = 6 ) back into the original dimensions:\n- Width = ( w = 6 )\n- Length = ( 2w = 12 )", "Now calculate the perimeter using the standard formula:\n[\n2 \ imes (6 + 12) = 2 \ imes 18 = 36\n]", "This matches the given perimeter, confirming the solution is correct.", "### Why This Equation Matters", "Understanding how to translate real-world descriptions into mathematical expressions—and solve them—is essential not only in geometry but in fields like architecture, construction, and design. This problem demonstrates how perimeter equations arise naturally when combining variables with standard formulas.", "### Summary", "- Perimeter of a rectangle: ( 2 \ imes (\ ext{length} + \ ext{width}) )\n- Given dimensions in terms of ( w ): width = ( w ), length = ( 2w )\n- Perimeter equation: ( 2(w + 2w) = 6w = 36 )\n- Solving yields ( w = 6 )\n- Verified perimeter: ( 36 ), consistent with problem statement", "Learning this process helps build strong problem-solving skills with geometric formulas. Whether you’re a student, teacher, or homeschooler, mastering perimeter calculations like this empowers you to tackle real-world shape problems with confidence.", "---\nKeywords: perimeter calculation, solving equations, geometry, rectangles formula, algebraic word problems, solve for ( w ), perimeter equals 36, geometry practice.", "Explore related topics to deepen your math skills:\n- How to use perimeter formulas for irregular shapes\n- Real-world applications of perimeter in everyday life\n- Step-by-step guide to solving linear equations like ( 6w = 36 )", "Start converting abstract variables into precise geometric solutions—one equation at a time."]