2.1 Scale
Scale is the ratio of a distance on the map to the real distance. On a 1 : 100,000 map, 1 cm stands for 100,000 cm, or 1 km. A scale with a small denominator is a "large scale": it shows a smaller area in more detail. A city plan might be 1 : 5,000 and a world map 1 : 50,000,000.
Careful with areas: if lengths are proportional to the scale, areas are proportional to its square. Enlarge a map twice and the same place covers 4 times the area on paper. Always make the units match first when converting: 1 km = 100,000 cm, 1 km² = 10¹⁰ cm².
FormulaReal distance = map distance × scale denominator · Real area = map area × (scale denominator)²
2.2 Contours and slope
Contour lines join points at the same height above sea level. The height difference between neighbouring lines is the contour interval. Where lines are close together the slope is steep; where they are far apart it is gentle. Closed rings inside each other show a hill — or a hollow if the inner ring is lower. In valleys contours make a "V" whose point faces uphill, towards the river's upper course; on ridges the point faces downhill.
Slope is the height difference between two points divided by the horizontal distance, usually given as a percentage. A road that rises 8 m over 100 m horizontally has an 8% gradient. In slope problems, the horizontal distance comes from the map scale and the height difference from the contours.
FormulaSlope (%) = height difference ÷ horizontal distance × 100
2.3 Projections
A sphere's surface can't be laid flat without tearing or stretching, so every map projection distorts something: area, shape, distance or direction. The Mercator projection preserves angles and local shapes; because constant-compass-bearing routes appear as straight lines, it was used in navigation for centuries. But it greatly exaggerates areas towards the poles.
On a Mercator map Greenland looks as big as Africa; in reality Africa is about 14 times larger. Equal-area projections show areas correctly but distort shapes. The right projection depends on the map's purpose: equal-area for population density, projections that show great circles for flight routes.
Key terms
- Large scale
- A scale with a small denominator, showing a small area in detail.
- Contour line
- A line joining points at the same height.
- Contour interval
- The height difference between neighbouring contours.
- Projection
- A method of transferring a globe's surface onto a flat map.
End-of-unit problems
Solve each one yourself first, then open the solution.
On a 1 : 250,000 map, two towns are 6.4 cm apart. What is the real distance in km? How many cm apart are they on a 1 : 1,000,000 map?
Show the solution
6.4 × 250,000 = 1,600,000 cm = 16 km. At 1 : 1,000,000, 16 km = 1,600,000 cm → 1,600,000 ÷ 1,000,000 = 1.6 cm.
On a 1 : 50,000 map, a lake covers 8 cm². What is its real area in km²?
Show the solution
1 cm on the map = 50,000 cm = 0.5 km, so 1 cm² = 0.5 × 0.5 = 0.25 km². 8 × 0.25 = 2 km². (The square of the scale is used; multiplying by the length factor alone would be wrong.)
On a contour map, point A lies on the 1250 m contour and B on the 750 m contour. On the 1 : 25,000 map they are 10 cm apart. What is the average slope in percent? Is it suitable for a walking path?
Show the solution
Height difference 500 m. Horizontal distance 10 × 25,000 = 250,000 cm = 2,500 m. Slope = 500 ÷ 2500 × 100 = 20%. An average of 20% is steep; a path should zigzag to reduce the gradient.
Looking at a Mercator map, a student says "Greenland is as big as Africa". Explain the mistake and say what kind of projection should be used to compare countries' sizes.
Show the solution
Mercator spaces the parallels farther apart towards the poles, so high-latitude areas look huge. Greenland lies between 60° and 83° N, while Africa straddles the Equator. In reality Africa is about 30 million km² and Greenland about 2.2 million km² (about 14 times smaller). To compare areas, an equal-area projection should be used.