Diagonal scale: hydropower dam-powerhouse

RF

1 km $= 1000$ m $= 10^5$ cm on the ground is represented by 25 cm.

$$\text{RF} = \frac{25}{10^{5}} = \frac{1}{4000}$$

Scale length

$$\text{Length for 500 m} = \frac{500 \times 100}{4000} = 12.5\ \text{cm}$$

Therefore 100 m $= 2.5$ cm, 10 m $= 0.25$ cm, 1 m $= 0.025$ cm.

Construction

1. Draw a line 12.5 cm long and divide it into 5 equal main divisions of 2.5 cm (100 m each). Add one more division of 2.5 cm to the left of the zero, so the total drawn is 15 cm. Number 0, 100, ... 500.
2. Divide the left extra division into 10 equal parts of 0.25 cm (10 m each).
3. Draw a rectangle of any convenient height (say 2.5 cm), divide the height into 10 parts and draw 10 horizontal lines.
4. Join the top of the first sub-division (10 m) with the bottom 0 of the main scale; draw parallel diagonals through the other sub-divisions. Each step along a diagonal equals 1 m.

 10|   /  /  /  /  /  /  /  /  /  /|
  5| /  /                          |
  0+---+----+----+----+----+----+--+
  90..0  100  200  300  400  500 (m)

Distances

• 123 m $= 100 + 20 + 3$: vertical at 100, the diagonal from the 20 m sub-division, 3rd horizontal line.
• 256 m $= 200 + 50 + 6$: vertical at 200, diagonal from 50 m, 6th horizontal line.

Answer: RF = 1/4000; length of scale = 12.5 cm (plus the left sub-divided part).