Surveying – Area from a reduced plan after proportional shrinkage Original plan scale = 10 m to 1 cm. A line 10 cm long becomes 9 cm after reduction. The reduced plan area is 81 cm². What is the actual area (m²) on the ground?

Introduction / Context:
When a plotted plan is photographically reduced or shrunk, linear dimensions scale by a factor r, and areas scale by r^2. Converting measured areas on such a reduced plan back to ground area requires careful handling of both the original plotting scale and the reduction factor.

Given Data / Assumptions:

• Original plotting scale: 10 m per 1 cm (i.e., 1 cm on plan represents 10 m on ground).
• After reduction, a line originally 10 cm now measures 9 cm → linear reduction factor r = 9/10.
• Area measured on reduced plan = 81 cm².
• Uniform shrinkage in both directions (isotropic scaling).

Concept / Approach:

First determine the effective scale of the reduced plan. If a ground length that was 100 m (since 10 cm × 10 m/cm) is now represented by 9 cm, then the reduced scale is 100 m / 9 cm. Ground area equals (plan area) * (scale in m/cm)². Alternatively, combine the original scale factor (10 m/cm) with the inverse of the reduction ratio (1 / r) to get the new scale: 10 / 0.9 = 11.111… m/cm, then square it and multiply by the reduced plan area.

Step-by-Step Solution:

Verification / Alternative check:

Using combined scale: new scale = 10 / 0.9 = 11.111… m/cm; area factor = 11.111…² = 123.456…; 81 × 123.456… = 10000 m², consistent with above.

Why Other Options Are Wrong:

6561 or 656 m² result from squaring 0.9 incorrectly or mixing centimetre and metre units; 1000 and 8100 m² ignore the correct squared scaling.

Common Pitfalls:

Forgetting that area scales with the square of the linear ratio, or applying the original scale without accounting for the additional photographic reduction.

Final Answer:

10000