Problem Statement
A plot of land acquired for a factory site measures 25 cm × 20 cm on a village map drawn to a scale of 1 cm = 100 m. What is its area in hectares? What will be its area on a toposheet with a scale of 1 : 50,000?
Step-by-Step Solution
Key Information
- Dimensions on Village Map = 25 cm × 20 cm
- Village Map Scale = 1 cm = 100 m
- Toposheet Scale = 1 : 50,000
- Goals: Find the true ground area (in hectares) and the area on the toposheet.
- Conversion: 1 hectare = 10,000 m²
Step 1: Calculate Ground Area from Village Map
First, determine the area represented by 1 cm² on the village map.
Linear Scale: 1 cm = 100 m
Area Scale: 1 cm² = (1 cm) × (1 cm) on map corresponds to (100 m) × (100 m) on ground.
Area Scale (Village Map) = 1 cm² = 10,000 m²
Next, calculate the area of the plot on the village map.
Area on Map = Length × Width = 25 cm × 20 cm
Area on Map = 500 cm²
Now, calculate the actual ground area using the map area and the area scale.
Ground Area = Area on Map (cm²) × Area Scale (m²/cm²)
Ground Area = 500 cm² × 10,000 m²/cm²
Ground Area = 5,000,000 m²
Finally, convert the ground area from square meters (m²) to hectares.
Ground Area (hectares) = Ground Area (m²) / 10,000 m²/hectare
Ground Area (hectares) = 5,000,000 m² / 10,000 m²/hectare
Ground Area = 500 hectares
Step 2: Calculate Area on Toposheet
Determine the linear and area scales for the toposheet.
Toposheet Scale = 1 : 50,000
This means 1 cm on the toposheet represents 50,000 cm on the ground.
Linear Scale: 1 cm = 50,000 cm = 500 m
Now find the area scale for the toposheet.
Area Scale: 1 cm² on toposheet = (500 m) × (500 m) on ground
Area Scale (Toposheet) = 1 cm² = 250,000 m²
Calculate the area the plot occupies on the toposheet using the ground area and the toposheet’s area scale.
Area on Toposheet (cm²) = Ground Area (m²) / Area Scale (m²/cm²)
Area on Toposheet = 5,000,000 m² / 250,000 m²/cm²
Area on Toposheet = 20 cm²
Final Results
The true area of the plot on the ground is 500 hectares.
The area of the plot on the 1:50,000 toposheet is 20 cm².
Conceptual Explanation & Applications
Core Concepts:
- Map Scale (Linear vs. Area): The linear scale (e.g., 1 cm = 100 m) relates map distance to ground distance. The area scale relates map area to ground area and is derived by squaring the linear scale factor (e.g., 1 cm² = (100 m)² = 10,000 m²).
- Area Calculation: Finding the space occupied by a shape. On a map, it’s calculated using map dimensions (length × width). On the ground, it’s calculated using the corresponding map area and the map’s area scale.
- Unit Conversion: Converting between different units of area, such as square meters (m²) and hectares (ha), is crucial for practical understanding (1 ha = 10,000 m²).
- Scale Transformation: Knowing the true ground area allows calculating its representation (area) on any map or plan, provided the scale of that map/plan is known.
Real-World Applications:
- Land Surveying & Real Estate: Determining accurate property sizes for valuation, development, and legal purposes.
- Urban and Regional Planning: Calculating land use areas (residential, commercial, parks) from master plans.
- Agriculture & Forestry: Estimating field sizes, calculating yields per area, managing forest plots.
- Civil Engineering & Construction: Site assessment, calculating material requirements (e.g., paving, turf) based on plan areas.
- Geographic Information Systems (GIS): Automated calculation of areas from digital maps for environmental studies, resource management, and infrastructure planning.
Why It Works:
The relationship between map and ground is defined by the scale. Since area involves two dimensions (length and width), the area scale factor is the square of the linear scale factor. By calculating the plot’s area on the initial map (500 cm²) and knowing its area scale (1 cm² = 10,000 m²), we find the true ground area (500 × 10,000 = 5,000,000 m²). Once the true ground area is known, we can reverse the process for any other map. For the toposheet (scale 1 cm = 500 m, so area scale 1 cm² = 250,000 m²), we divide the ground area by the toposheet’s area scale (5,000,000 m² / 250,000 m²/cm²) to find how much area the plot occupies on that specific map (20 cm²). This demonstrates how scale dictates the representation of real-world features on different maps.
