Problem #56 MEDIUM
The Landlord's Garden
Scenario Geometry Logic Math
Problem Statement
Suresh owns a rectangular piece of land in Coimbatore that measures 60 metres long and 40 metres wide. He wants to create a garden path that runs diagonally across his property — from the top-left corner to the bottom-right corner.
His gardener, Murugan, needs to know the exact length of this diagonal path so he can buy the right amount of paving stones. Each paving stone is 1 metre long.
'Just measure it,' Suresh says.
'I cannot,' Murugan says. 'The land is full of trees and I cannot walk in a straight line across it. But I know the two side lengths.'
Suresh pulls out a scrap of paper. He sketches a rectangle and draws a diagonal line across it.
'60 metres by 40 metres,' Suresh says. 'We can calculate the exact length without measuring it at all.'
What is the exact length of the diagonal path across Suresh's 60 x 40 metre rectangular land? How many paving stones does Murugan need to buy, and why might he need to buy one extra?
Answer & Quick Explanation
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The diagonal is exactly 20√13 ≈ 72.11 metres. Murugan needs to buy 73 paving stones — 72 for the full metres and 1 extra for the remaining 0.11 metres, since stones cannot be purchased in fractions.
Detailed Editorial Solution
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The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. A rectangle's diagonal is the hypotenuse of a right triangle formed by the two sides.
Step 1: The rectangle has length 60 m and width 40 m. Draw the diagonal — it forms a right triangle with legs of 60 m and 40 m.
Step 2: Apply Pythagoras: D² = 60² + 40² = 3600 + 1600 = 5200.
Step 3: D = √5200. Simplify: 5200 = 400 x 13. So √5200 = √400 x √13 = 20√13.
Step 4: Calculate numerically: √13 ≈ 3.6056. So D ≈ 20 x 3.6056 = 72.11 metres.
Step 5: Murugan needs 72.11 metres of paving stones. Since stones come in 1-metre units, he must buy 73 stones (you cannot buy 0.11 of a stone — the partial metre still needs a full stone).
Step 6: The exact answer is 20√13 metres. The practical answer is 73 stones.
Key Insight:
The Pythagorean theorem is the foundation of all distance calculations in flat 2D space. The gap between the exact answer (20√13) and the practical answer (73 stones) illustrates a real-world consideration that pure mathematics ignores: physical materials come in discrete units, so always round up when buying materials.