Tue, 05 May 2026 13:15:38 +0000

A rectangular garden is 20 meters long and 10 meters wide. A path of uniform width is to be built around the garden, increasing the total area to 396 square meters. What is the width of the path?
This question reflects a growing interest in smart outdoor space optimization across U.S. home gardens. With rising urban living, many homeowners seek ways to expand usable areas without sacrificing green space—making conceptual layouts like this increasingly relevant. The symmetry and simplicity of the starting rectangle lend themselves naturally to mathematical exploration, especially when area expansion is involved. The rectangular garden’s original area is 200 square meters (20 × 10). Adding a path uniformly surrounds it, increasing both length and width by twice the path’s width. If the path width is *x* meters, the new overall dimensions become (20 + 2x) meters in length and (10 + 2x) meters in width. Area calculation follows the formula:
Total area = (20 + 2x)(10 + 2x)
Set this equal to 396:
(20 + 2x)(10 + 2x) = 396
Expanding:
200 + 40x + 20x + 4x² = 396
Simplifies to:
4x² + 60x + 200 = 396
4x² + 60x – 196 = 0
Divide equation by 4:
x² + 15x – 49 = 0 Solving this quadratic equation using the quadratic formula, x = [–15 ± √(225 + 196)] / 2 = [–15 ± √421] / 2
The positive root is x ≈ (–15 + 20.52) / 2 ≈ 2.76 meters.
Testing this value shows the new area hovers near 396 m², confirming practicality. This practical calculation matters as homeowners balance functionality with aesthetics. Even small width increments—around 2.75 meters—can significantly transform how outdoor space functions.