A rectangular garden measuring 12 meters by 8 meters is surrounded by a path of uniform width. The total area of the garden and the path is 200 square meters. What is the width of the path?

["Title: How to Calculate the Uniform Path Width Surrounding a Rectangular Garden – Solved!", "Creating a beautiful, uniform rectangular garden with a path around it offers both aesthetic appeal and functional space. In one common landscaping challenge, a 12m × 8m garden is enclosed by a path of equal width on all sides, and the total area—including the path—is 200 square meters. This article walks you through the step-by-step solution to find the width of the path, demonstrating how to apply algebra and geometry for real-world garden design.", "---", "### Understanding the Problem", "You have a garden measuring 12 meters by 8 meters, surrounded by a path of uniform width. The entire area—garden plus path—is 200 square meters. We need to determine the width of the path.", "Let’s denote:\n- Garden dimensions: width = 8 m, length = 12 m\n- Path width (uniform around all sides): x meters", "---", "### Step 1: Calculate the Total Dimensions Including the Path", "Since the path surrounds the garden evenly:\n- Total length of garden + path = 12 + 2x\n- Total width of garden + path = 8 + 2x", "Thus, the total area (garden plus path) is:\n[\n(12 + 2x)(8 + 2x) = 200\n]", "---", "### Step 2: Expand and Simplify the Equation", "Expand the left-hand side:\n[\n(12 + 2x)(8 + 2x) = 12 \cdot 8 + 12 \cdot 2x + 2x \cdot 8 + 2x \cdot 2x = 96 + 24x + 16x + 4x^2\n]\n[\n= 4x^2 + 40x + 96\n]", "Set equal to total area:\n[\n4x^2 + 40x + 96 = 200\n]", "Subtract 200 from both sides:\n[\n4x^2 + 40x + 96 - 200 = 0\n]\n[\n4x^2 + 40x - 104 = 0\n]", "---", "### Step 3: Simplify the Quadratic Equation", "Divide every term by 4 to simplify:\n[\nx^2 + 10x - 26 = 0\n]", "---", "### Step 4: Solve the Quadratic Using the Quadratic Formula", "For any quadratic equation ( ax^2 + bx + c = 0 ), the solution is:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 10 ), ( c = -26 ):\n[\nx = \frac{-10 \pm \sqrt{10^2 - 4(1)(-26)}}{2(1)} = \frac{-10 \pm \sqrt{100 + 104}}{2} = \frac{-10 \pm \sqrt{204}}{2}\n]", "Simplify ( \sqrt{204} ):\n[\n\sqrt{204} = \sqrt{4 \ imes 51} = 2\sqrt{51} \approx 14.28\n]", "So:\n[\nx = \frac{-10 \pm 2\sqrt{51}}{2} = -5 \pm \sqrt{51}\n]", "Only the positive solution makes sense physically:\n[\nx = -5 + \sqrt{51}\n]", "Estimate:\n[\n\sqrt{51} \approx 7.14 \Rightarrow x \approx -5 + 7.14 = 2.14\n]", "But for exact form, we keep:\n[\nx = -5 + \sqrt{51}\n]", "---", "### Step 5: Interpret the Result", "The width of the path is ( \boxed{-5 + \sqrt{51}} ) meters — approximately 2.14 meters.", "However, since this is a real-world garden design, we check if this value yields a reasonable area:", "- Total length: 12 + 2(–5 + √51) ≈ 12 + 2(2.14) = 16.28 m\n- Total width: 8 + 2(–5 + √51) ≈ 8 + 4.28 = 12.28 m\n- Area: 16.28 × 12.28 ≈ 200.0 m² ✓", "Thus, the exact width is:\n[\n\boxed{-5 + \sqrt{51}} \ ext{ meters}\n]", "For practical purposes, this is about 2.14 meters, but the precise mathematical answer maintains exactness.", "---", "### Why This Calculation Matters", "Understanding how to compute path widths in rectangular garden designs helps homeowners and landscape architects:\n- Optimize space usage\n- Plan walkways comfortably\n- Estimate materials for paving, soil, and plants\n- Visualize final design before construction", "---", "### Final Thoughts", "Solving for the uniform path width around a garden isn’t just theoretical—it’s a practical tool for creating functional, elegant outdoor spaces. With a simple algebraic setup, even complex garden surrounds can be precisely measured and improved.", "Pro tip: Use this method whenever designing symmetrical garden paths—whether paved, gravel, or green lawn!", "---", "Keywords: rectangular garden with path, path width calculation, rectangles and areas, landscaping math, garden design solutions, algebra for gardens, uniform sidewalk width, garden planning total area.\nMeta description: Learn how to find the uniform path width around a 12m × 8m garden with total area 200 m²—solve with algebra and geometry for perfect landscaping design."]