Question

How do I calculate the scale of a map when 12km is represented by 4cm and determine the actual area of a field represented by 32cm^2 on the map?

Answer 1

To calculate the scale of the map, you start by determining the ratio of distance on the map to the actual distance. In this case, 12 km corresponds to 4 cm.

1. **Convert kilometers to centimeters**: Since 1 km equals 100,000 cm, we convert 12 km to centimeters:
\[
12 \, \text{km} = 12 \times 100,000 \, \text{cm} = 1,200,000 \, \text{cm}
\]

2. **Determine the scale**: Now, we can calculate the scale of the map. The scale is given by the ratio of the distance on the map to the actual distance:
\[
\text{Scale} = \frac{\text{Map Distance}}{\text{Actual Distance}} = \frac{4 \, \text{cm}}{1,200,000 \, \text{cm}} = \frac{1}{300,000}
\]
So the scale of the map is 1:300,000.

Next, to determine the actual area of a field represented by 32 cm² on the map:

1. **Calculate the actual area**: Since area is a two-dimensional measure, you need to square the scale factor when converting the area from the map to the actual area.

- Area scale factor = \(300,000 \times 300,000 = 90,000,000,000\).

2. **Convert the area**:
\[
\text{Actual Area} = \text{Map Area} \times \text{Area Scale Factor} = 32 \, \text{cm}^2 \times 90,000,000,000
\]
\[
\text{Actual Area} = 2,880,000,000 \, \text{cm}^2
\]

3. **Convert cm² to km²**: Since \(1 \, \text{km}^2 = 10,000,000 \, \text{cm}^2\):
\[
\text{Actual Area in km}^2 = \frac{2,880,000,000 \, \text{cm}^2}{10,000,000 \, \text{cm}^2/\text{km}^2} = 288 \, \text{km}^2
\]

Therefore, the actual area of the field represented by 32 cm² on the map is **288 km²**.