A rectangular field has a length of 120 meters and a width of 80 meters. A path of uniform width is built around the field, increasing the total area to 13,000 square meters. What is the width of the path?

Wed, 06 May 2026 02:40:34 +0000

A rectangular field has a length of 120 meters and a width of 80 meters. A path of uniform width is built around it, expanding the total area to 13,000 square meters. What is the width of the path? This question reflects growing interest in spatial design, sustainable landscaping, and functional outdoor spaces—trends amplified by rising home renovation engagement and smart land use in the U.S. Many homeowners and planners are reimagining traditional yards to balance aesthetics with practicality, especially in suburban settings where outdoor areas often define lifestyle value. A rectangular field with dimensions 120m × 80m covers 9,600 square meters. When a uniform-footpath surrounds the field, it adds consistent border space, increasing total area to 13,000 square meters—an expansion of 1,400 square meters. This precise geometry makes the problem both accessible for calculation and relevant to real-life design challenges facing U.S. property owners today. The field’s original area is 120 × 80 = 9,600 m². The expanded area with path adds 1,400 m², forming a new rectangle with outer dimensions based on uniform widths on each side. If the path width is *x* meters, then the total length becomes (120 + 2*x) and width becomes (80 + 2*x). Multiplying these gives:
(120 + 2x)(80 + 2x) = 13,000
Expanding:
9,600 + 400x + 4x² = 13,000
Simplifying:
4x² + 400x – 3,400 = 0
Dividing through by 4:
x² + 100x – 850 = 0 This quadratic equation models the true challenge behind measuring and optimizing outdoor space, showing how small uniform expansions can yield substantial changes. Solving this gives x ≈ 6 meters—an outward expansion safe enough in most residential contexts but meaningful in area impact.