A rectangular plot of land is 30 meters long and 20 meters wide. A path of uniform width is constructed around the plot, increasing the total area to 924 square meters. What is the width of the path? - Coaching Toolbox

Understanding the Context

Why a Rectangular Plot and a Uniform Path Are Trending Now

Across American suburban and urban neighborhoods, landowners and planners are seeking smarter ways to maximize outdoor functionality without sacrificing aesthetics. The fixed dimensions—30 meters (approximately 98 feet) by 20 meters (65 feet)—represent a practical, accessible starting point widely recognized in architectural and landscaping references. Around such plots, construction of a consistent gravel, stone, or plant-lined path is a common enhancement, not just a decorative choice. What’s driving this popularity? Economic shifts, rising interest in green spaces, and a desire for functional, low-maintenance outdoor areas are reshaping how homeowners visualize their plots. A path of uniform width doesn’t just separate; it defines, elevates, and increases usable space—making the math behind its width a key design variable.

Breaking Down the Math: Finding the Path Width

Key Insights

Calculating the uniform width of the surrounding path begins with a straightforward extension of rectangles. The original plot spans 30 meters by 20 meters. With a path of equal width, x, added evenly along all four sides, the total length and width grow by 2x:

New length: 30 + 2x
New width: 20 + 2x
New area: (30 + 2x)(20 + 2x) = 924 square meters

Expanding this equation gives:
600 + 100x + 4x² = 924
Simplifying:
4x² + 100x – 324 = 0

Using the quadratic formula:
x = [–100 ± √(100² – 4×4×(–324))]/(2×4)
x = [–100 ± √(10000 + 5184)]/8
x = [–100 ± √(15184)]/8
√15184 ≈ 123.22
x ≈ (–100 + 123.22)/8 ≈ 23.22/8 ≈ 2.9 meters

So, a path width of roughly 2.9 meters increases the area from 600 to 924, perfectly matching the target. This precise figure reflects both accuracy and practical design compromise.

Final Thoughts

Common Queries and How to Approach Each

H3: Why calculate the exact path width instead of estimating?
Estimates ignore critical differences in square footage. Even a 0.5-meter variance changes area significantly—sometimes by over