Measuring Distances and Areas

1How far, and how big?

A district officer planning a new health post wants to know how far the nearest families must walk; a tea company wants the area of its estate. Both questions are answered from the map with a ruler, a piece of string or paper, and the scale. This lesson uses Map A again.

2Straight-line distances

Lay the edge of a strip of paper between the two points, mark both on it, and lay the strip along the bar scale. If the distance is longer than the bar, read the whole bar, move the strip and add on. On a gridded map you can also calculate the distance from the grid references: the eastings give the distance east or west, the northings the distance north or south, and Pythagoras' theorem gives the straight line. This is a check, not a replacement, for measuring.

3Curved distances

Roads, rivers and railways bend, so a ruler cannot measure them directly. Two methods are used:

• Paper edge. Put the straight edge of a strip at the start (A) and mark it. Pivot the strip at each bend so that its edge follows the next nearly straight section, marking each point (B, C, D…) as you go. The distance from the first to the last mark is the length of the road. Use more, shorter sections on tight bends.
• String or thread. Lay a thread along the road, following every bend, mark the end point on it, then straighten it along the bar scale.

Both methods slightly underestimate a winding route, because small bends are smoothed out. Give the answer to one decimal place of a kilometre and say "about".

4Areas of regular shapes

A rectangle, square or triangle is measured with a ruler and the usual formulas. The tea estate on Map A is a rectangle from easting 28.5 to 29.5 and from northing 43.0 to 44.5:

$$\text{area}=1.0\text{ km}\times 1.5\text{ km}=1.5{\text{km}}^2=150\text{ ha}\mathrm{.}\backslash text\{area\}\; =\; 1.0\backslash \; \backslash text\{km\}\; \backslash times\; 1.5\backslash \; \backslash text\{km\}\; =\; 1.5\backslash \; \backslash text\{km\}^2\; =\; 150\backslash \; \backslash text\{ha\}.$$

Scale and area. Areas do not shrink in the same ratio as lengths. At 1:50 000, 1 cm on the map is 0.5 km, so 1 cm² on the map is $0.5\times 0.5=0.250.5\; \backslash times\; 0.5\; =\; 0.25$ km². The tea estate, 1.5 km², covers $\frac{1.5}{0.25}=6\backslash frac\{1.5\}\{0.25\}\; =\; 6$ cm² of the original sheet.

5Areas of irregular shapes: counting squares

For a forest, a lake or a swamp, count grid squares:

1. Count every square that is completely covered.
2. Count each square that is more than half covered as a whole square, and ignore squares less than half covered. (A second method counts all part-covered squares and halves their number.)
3. Multiply by the area of one square (1 km² on Map A).

In Figure 4 no square is completely covered; three squares are more than half covered (0.75, 0.97 and 0.61). So the forest is about 3 km². The second method gives $\frac{6}{2}=3\backslash frac\{6\}\{2\}\; =\; 3$ km² as well. The true area of the shape is 2.94 km², so the estimate is good. For a small shape, lay a transparent sheet ruled in 2 mm squares over the map: each 2 mm square at 1:50 000 is $0.1\times 0.1=0.010.1\; \backslash times\; 0.1\; =\; 0.01$ km², or 1 ha, so the count is in hectares.