SQA Advanced Higher Geography

What the scale means

The Ordnance Survey Explorer series is drawn at 1:25,000, so one unit on the map stands for 25,000 of the same unit on the ground. This is a representative fraction. Working in centimetres, 1 cm on the map is 25,000 cm on the ground, which is 250 m. The scale is therefore usually remembered as 4 cm to 1 km, because 4 cm is exactly 1 km.

Every map carries a linear scale (a scale bar) as well as the representative fraction. The bar is the safer thing to measure against when a map has been photocopied or reduced, because a bar shrinks with the page and the fraction does not. On a reduced copy the printed grid squares are less than 4 cm across, which shows at once that the fraction no longer applies.

A larger scale shows more detail over a smaller area. A 1:25,000 map is a larger scale than a 1:50,000 map, so it shows individual field boundaries, rights of way and small buildings that the 1:50,000 map leaves out or combines.

Straight-line distance

For a straight line, measure with a ruler from one point to the other and convert.

1. Measure the map distance in centimetres.
2. Multiply by 250 to give metres.
3. Divide by 1,000 if the answer is wanted in kilometres.

For example, two farms are 7.3 cm apart on the map. The ground distance is $7.3 \times 250 = 1\,825$ m, or 1.8 km to the nearest hundred metres.

Where the line is long, count whole grid squares along it instead, since each square is 1 km across, and then measure the remainder. A diagonal across one grid square is longer than 1 km: it is $\sqrt{2}$ km, about 1.4 km.

Winding routes

Roads, rivers and paths curve, so a ruler gives an underestimate. Two methods work well.

1. Lay the straight edge of a piece of paper along the route and mark the start. Pivot the paper at each bend, keeping the edge on the line, and mark the point where the route leaves the paper. Carry on until the finish, then hold the paper against the scale bar and read off the total.
2. Lay a piece of string or thread along every bend, mark the finish, then straighten the string and measure it against a ruler or the scale bar.

Both methods come out slightly short of the true length, because small bends are smoothed over. Both measure a horizontal distance, so a route up a steep hillside is longer on the ground than on the map. The effect is small for modest gradients and only matters on very steep slopes.

A route must often fall in a stated range, for example between 17 and 20 km for a cycle race. At 1:25,000 that is a length of between 68 cm and 80 cm of string, because $17\,000 \div 250 = 68$ and $20\,000 \div 250 = 80$. Working out this centimetre range first saves measuring repeatedly.

Grid squares and area

The blue grid lines on an Explorer map are 1 km apart, so each grid square is 4 cm by 4 cm on the map and covers 1 square kilometre, written 1 km², on the ground. The area of a lake, wood, urban area or farm can be estimated from this.

1. Count the grid squares that are completely covered.
2. Count the squares that are partly covered.
3. Treat each partly covered square as a half square, or count squares that are more than half covered as whole and the others as nothing.
4. Add the totals.

A reservoir that covers 11 whole squares and 9 part squares has an area of about $11 + 9 \times 0.5 = 15.5$ km². Counting can be made more accurate by tracing the outline on to tracing paper laid over a finer grid, in which every 2 mm by 2 mm square on the map is 0.25 ha.

For a regular shape, the area is calculated directly. A rectangular field measuring 3.2 cm by 2.0 cm on the map is $800 \text{ m} \times 500 \text{ m}$ on the ground, so its area is $400\,000$ m², which is 40 hectares because one hectare is $10\,000$ m².

An area changes with the square of the length. Halving every length on a map cuts every area to a quarter, so areas cannot be compared between maps at different scales without converting both to ground units.

Drawing to scale

A question may ask for a distance, route or shape to be drawn to scale. The scale is used in reverse. To mark a point 1.6 km from a church on a bearing, convert 1.6 km to centimetres on the map, $1\,600 \div 250 = 6.4$ cm, and measure that distance with a ruler. A route of a stated ground length is drawn as the matching length of line on a tracing overlay. A required length is checked against the centimetre range worked out in advance.

The same conversion helps with sketch maps and cross sections, where a horizontal scale has to be chosen so that the finished diagram fits a page. A 5 km section drawn at 1:25,000 is 20 cm long and fits an A4 page.

Scale on other items

A photograph rarely has a single scale because of perspective: features in the foreground are larger than the same features in the background, so distances are best taken from the map. A sketch has no scale unless one is added. A diagram that has been enlarged or reduced stays correct if it has a bar scale, which is why the bar is the one to use.

Gradient, area, density and journey time all depend on correct distances. A density worked out from an area counted in grid squares is only as accurate as the count, and a quoted density gives its units, such as people per km².