Scale and area on topo sheet

Scale of the plan

Map area $= 8 \times 9 = 72$ cm$^2$. Ground area $= 18000 \times 10^{4} = 1.8 \times 10^{8}$ cm$^2$.

$$\text{RF}^2 = \frac{72}{1.8 \times 10^{8}} = 4.0 \times 10^{-7}$$ $$\text{RF} = 6.3246 \times 10^{-4} = \frac{1}{1581.1}$$

So 1 cm on the plan $= 15.81$ m on the ground (approx. 1:1581).

Sketch of a scale reading 1 m

Take a maximum reading of 150 m: length $= 150 \times 100/1581.1 = 9.49$ cm (100 m $= 6.32$ cm, 10 m $= 0.632$ cm, 1 m $= 0.0632$ cm).

1. Draw a line 9.49 cm long and divide it into main divisions of 50 m ($3.16$ cm each): three divisions for 150 m. Add one more division to the left of 0.
2. Divide the left main division into 5 sub-divisions of 10 m each ($0.632$ cm).
3. Draw 10 equally spaced horizontal lines on a rectangle about 2 cm high, and draw diagonals from the top of each sub-division to the next lower sub-division on the base. Each step on a diagonal reads 1 m (a diagonal scale is needed because 1 m is only 0.63 mm).

 10 |  /  /  /  /  /  /  /  /  /  |
  5 | /  /                        |
  0 +---+-----+-----+-------------+
   50..0   50   100   150 (m)

Area on the 1:12500 sheet

Linear ratio $= 1/12500$, so the same ground area shown on the topo sheet is

$$\text{Area on sheet} = \frac{18000 \times 10^{4}\ \text{cm}^2}{12500^{2}} = 1.152\ \text{cm}^2$$

The plan would measure about $1.01\ \text{cm} \times 1.14\ \text{cm}$ on the sheet (each side of 8 cm and 9 cm is reduced by the factor $1581.1/12500$).

Answer: RF $\approx$ 1/1581 (1 cm = 15.81 m); area on the 1:12500 sheet = 1.152 cm$^2$.