A man walks diagonally across a square instead of along two edges. Approximately what percentage distance is saved?

Problem restatementCompare the diagonal path to the L-shaped edge path around one corner of a square and find the percentage saving in distance.

Given data

• Square side = a.
• Edge path distance = a + a = 2a.
• Diagonal distance = a√2.

Concept/ApproachPercent saved = [(edge path − diagonal) ÷ edge path] × 100%.

Step-by-Step calculationPercent saved = [(2a − a√2) ÷ 2a] × 100%= (1 − √2 ÷ 2) × 100% ≈ (1 − 0.7071) × 100%≈ 29.3%

Verification/AlternativeFor a = 1: edge = 2, diagonal ≈ 1.4142, saving ≈ 0.5858 ⇒ 0.5858 ÷ 2 ≈ 29.29%.

Common pitfallsUsing 1 − (1/√2) instead of 1 − (√2/2); both are equal numerically here, but ensure the reference distance is the edge path (2a).

Final Answer29.3%