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? - Treasure Valley Movers

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.

Why is this problem gaining traction now? Home improvement platforms and gardening influencers are highlighting space-saving design solutions, especially as many U.S. families adapt to compact yet inviting outdoor environments. The 20x10 layout is ideal for standard homes, offering a proven benchmark for real-world planning.

For those facing similar layouts, the rectangular plan allows predictable math and smooth materials planning—essential for both DIYers and contractors. Understanding the expansion math empowers better decision-making beyond just aesthetics: materials, maintenance, and usability all benefit from precise dimensions.

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