Diagonal scale: Pokhara-Kathmandu 200 km

RF

Ground distance = 200 km $= 200 \times 10^5$ cm $= 2 \times 10^7$ cm; map distance = 5 cm.

$$\text{RF} = \frac{5}{2 \times 10^7} = \frac{1}{4{,}000{,}000}$$

Length of the diagonal scale

$$\text{Length} = \frac{600\ \text{km} \times 10^5\ \text{cm}}{4 \times 10^6} = 15\ \text{cm}$$

1 km on the scale $= 10^5/(4\times10^6) = 0.025$ cm; 10 km $= 0.25$ cm; 100 km $= 2.5$ cm.

Construction

1. Draw a line 15 cm long and divide it into 6 equal parts of 2.5 cm; each represents 100 km. Number them 0, 100, ... 500 starting from the end of the first division (the first, left-hand division is the one that is sub-divided).
2. Divide the left-most main division into 10 equal parts of 0.25 cm; each is 10 km. Number them 10, 20, ... 90 to the left of 0.
3. At the ends, erect perpendiculars of height 2.5 cm (any convenient height) and divide each into 10 equal parts; draw horizontal lines through these points. Complete the rectangle.
4. Join the top of the first sub-division (10 km) to the 0 on the bottom line, and draw diagonals parallel to it through each of the other sub-divisions. Each horizontal step along a diagonal is 1 km (0.025 cm).

 10 |    /  /  /  /  /  /  /  /  /  /|
  8 |   /  /  /                        |
  6 |  /  /                            |
  4 | /  /                             |
  2 |/                                 |
  0 +---+---+---+---+---+---+---+---+---+
   90..10 0   100   200   300   400   500  (km)

Indicating the distances

• (i) 222 km $= 200 + 20 + 2$: take the vertical line at 200, the diagonal that starts at the 20 km sub-division, and the 2nd horizontal line. Mark the point where this diagonal meets the 2nd horizontal line; the length along that line from this point to the vertical at 200 is 222 km.
• (ii) 336 km $= 300 + 30 + 6$: use the vertical at 300, the diagonal from the 30 km sub-division and the 6th horizontal line.

Answer: RF = 1/4,000,000; scale length = 15 cm.