A high school student is designing a garden with a circular pond and a semicircular walkway around it. If the radius of the pond is \(r\) units and the radius of the walkway is \(5r\) units, what is the ratio of the area of the pond to the area of the walkway?

["### Factoring Nature Into Science: How a High School Student’s Circular Garden Features a Perfect Pond and Walkway Ratio", "Designing a garden isn’t just an artistic endeavor—it’s a practical science project. For a high school student exploring geometry and sustainability, integrating a circular pond with a surrounding semicircular walkway offers a real-world way to apply mathematical concepts. If the pond has radius ( r ) and the outer radius of the semicircular walkway is ( 5r ), understanding the area ratio between the pond and the walkway reveals how thoughtful landscaping can maximize space and beauty.", "This article breaks down the geometry behind the design, calculates the area ratio, and explains why such a concept matters beyond aesthetics.", "---", "### Understanding the Garden Layout", "The garden features two key circular elements:\n- A central pond with radius ( r ).\n- A surrounding walkway that wraps around the pond in a semicircular shape, extending outward to a total outer radius of ( 5r ).", "Importantly, the walkway isn’t a full disc—but a semicircular ring. This design optimizes space and creates a visually appealing pathway that plays evenly with the pond’s circular form.", "---", "### Step 1: Calculate the Area of the Pond", "The pond is a full circle, so its area is straightforward:\n[\n\ ext{Area}{\ ext{pond}} = \pi r^2\n]", "---", "### Step 2: Calculate the Area of the Walkway", "The walkway forms a semicircular ring around the pond. To find its area:\n1. The outer semicircular radius is given as ( 5r ).\n2. The inner radius (blocking off the pond) is ( r ).", "But since the walkway only exists as a semicircular annular region, we compute the area between the outer and inner semicircles:\n[\n\ ext{Area} \pi r^2}} = \frac{1}{2} \pi (5r)^2 - \frac{1}{2\n]", "Simplify:\n[\n\ ext{Area}{\ ext{walkway}} = \frac{1}{2} \pi \left(25r^2 - r^2\right) = \frac{1}{2} \pi (24r^2) = 12\pi r^2\n]", "---", "### Step 3: Compute the Area Ratio", "The ratio of the pond’s area to the walkway’s area is:\n[\n\ ext{Ratio} = \frac{\ ext{Area}}}}{\ ext{Area}_{\ ext{walkway}}} = \frac{\pi r^2}{12\pi r^2\n]", "The ( \pi r^2 ) terms cancel out:\n[\n\ ext{Ratio} = \frac{1}{12}\n]", "---", "### Why This Ratio Matters", "This 1:12 ratio reflects a thoughtful balance:\n- The small pond anchors the design, promoting biodiversity (a habitat for frogs, insects) and reflection.\n- The larger semicircular walkway offers a graceful, accessible space for walking and contemplation—encouraging outdoor engagement.\n- Geometrically, it shows how circular symmetry supports both function and beauty in eco-friendly landscaping.", "For high school students, understanding such ratios connects classroom math to real-world design, sparking creativity and scientific awareness.", "---", "### Conclusion", "By designing a garden with a circular pond and semicircular walkway, students apply key geometric principles to create meaningful outdoor spaces. The ratio of the pond’s area to the walkway’s area—1:12—is more than a calculation; it’s a symbol of harmony between natural form and intentional design.", "In the ever-growing movement toward sustainable education and green living, projects like this remind us that learning isn’t confined to textbooks—it grows in every garden bed.", "---", "Keywords: high school garden design, circular pond ratio, semicircular walkway area, geometry in landscaping, circular area calculation, student-led sustainability project, math and nature integration, pond to walkway ratio."]