Der Umfang ist \( 2(w + 2w) = 36 \) → \( 2(3w) = 36 \) → \( 6w = 36 \) → \( w = 6 \).

["Understanding and Solving the Equation: Der Umfang ist ( 2(w + 2w) = 36 )", "In geometry, solving equations to find unknown dimensions is a fundamental skill, especially when dealing with perimeter problems. One classic example is finding the value of ( w ) in the equation ( 2(w + 2w) = 36 ), commonly used to determine side lengths from perimeter values.", "### Breaking Down the Equation", "Let’s walk through solving this step-by-step:", "1. Simplify inside the parentheses\n The expression ( w + 2w ) combines the variable ( w ) with twice itself:\n [\n w + 2w = 3w\n ]", "2. Apply the multiplication outside\n The factor 2 multiplies the entire sum:\n [\n 2(3w) = 36\n ]", "3. Simplify the left side\n Multiplying 2 by 3w gives:\n [\n 6w = 36\n ]", "4. Solve for ( w )\n Divide both sides by 6:\n [\n w = \frac{36}{6} = 6\n ]", "### Applying the Result to Real Geometry", "With ( w = 6 ), we return to the geometric context:\nThis equation likely represents the perimeter of a rectangle where one side is ( w ) and the adjacent side is ( 2w ). Therefore, the full perimeter is:\n[\n\ ext{Perimeter} = 2(w + 2w) = 2(3w) = 6w\n]", "Set this equal to the given perimeter of 36 units:\n[\n6w = 36\n]", "Solving gives ( w = 6 ), meaning:\n- One side is 6 units\n- The adjacent side is ( 2w = 12 ) units", "This confirms a rectangle with consistent proportions: ( 6 \ imes 12 ), correctly yielding a perimeter of ( 2(6 + 12) = 36 ).", "### Why This Equation Matters in Geometry", "Der Umfang (perimeter) calculation is not only useful for basic shapes but also foundational in design, construction, and spatial reasoning. Mastering such algebra ensures accuracy and confidence in solving real-world measurement problems.", "### Final Answer", "[\n\boxed{w = 6}\n]", "Use this step-by-step approach whenever you encounter perimeter equations involving linear expressions—simplify, simplify, solve—to find key dimensions with precision."]