Errors in Surveying MCQ with answer and solution

Concept

Over small distances the curved earth surface can be treated as flat because the arc is almost equal to its chord.

Formula

Arc - Chord ≈ L³ / (24 R²)

Step-by-step solution

1. Arc length L = 12 km, earth radius R ≈ 6370 km. 2. Difference = L³ / (24 R²). 3. L³ = 12³ = 1728 km³. 4. 24 R² = 24 × 6370² = 24 × 40576900 = 973845600 km². 5. Difference = 1728 / 973845600 = 1.77 × 10^-6 km. 6. 1.77 × 10^-6 km = 1.77 mm ≈ 0.18 cm. 7. The nearest option is 0.1 cm.

Explanation

In plane surveying we ignore the curvature of the earth. For a 12 km arc, the arc is longer than its chord by only about 0.18 cm, which is very small. So the earth surface can be treated as a plane for small areas.

Why the other options are wrong

B (10 cm) and C (100 cm) are far too large. D (1 cm) is a common textbook statement, but calculation gives only about 0.18 cm.

Shortcut

Use Arc - Chord = L³/(24R²) with L and R in km, then convert the result to cm.

Common mistake

Using L² instead of L³, or making a mistake in unit conversion from km to cm.