Scale drawings and maps

On a 1:24,000 map, 1 inch stands for 24,000 inches on the ground, which is exactly 2,000 feet.

Is a 1:100,000 map a larger scale than 1:24,000?

A bigger number feels like a bigger scale. It is the other way round. A 1:100,000 map shrinks the ground 100,000 times and a 1:24,000 map only 24,000 times. A scale is a fraction, and 1/100,000 is the smaller one. A second wrong idea is that every map is drawn to scale. One exception is the subway route map. A 2016 post on the Korean Ministry of Education's official blog compares the route map of Seoul's Line 1 with the real line and notes that direction, position and distance all differ. A route map shows the order of stations and where to change. A ruler on it tells you nothing about kilometres.

Map distance compared with ground distance

No sheet of paper is as big as a town, so every map shrinks the ground by the same factor. The USGS fact sheet "Map Scales" defines scale as the relationship between distance on the map and distance on the ground, and says it is usually given as a fraction or ratio, such as 1/10,000 or 1:10,000. That means 1 unit on the map, 1 inch or 1 centimetre, stands for 10,000 of the same units on the ground. The first number is always 1.

Because both sides of the ratio use the same unit, a scale works with any unit you like. On a 1:50,000 map, 1 cm is 50,000 cm on the ground. Change that to metres by dividing by 100: 500 m. So 1 cm on the map is half a kilometre, which matches the USGS table for 1:50,000 maps.

4.2 cm on the map, 2.1 km on the ground

Map to ground: multiply by the scale number. Two villages are 4.2 cm apart on a 1:50,000 map. 4.2 × 50,000 = 210,000 cm, which is 2,100 m or 2.1 km. Ground to map: divide. A 3 km trail on a 1:25,000 map is 300,000 cm ÷ 25,000 = 12 cm long. Change to one unit first. Mixing centimetres and kilometres in one step is an easy way to be off by a factor of 100,000.

The same idea runs plans and models. A made-up floor plan drawn at 1:50 shows a room 8 cm long, so the real room is 8 × 50 = 400 cm, or 4 m. A model car at 1:24 that is 18 cm long stands for a car 18 × 24 = 432 cm long. And a made-up street map beside a station exit, marked 1:5,000, puts a library 6 cm away at 6 × 5,000 = 30,000 cm, or 300 m. A ruler only works on a map that states its scale.

Bigger second number, smaller scale

USGS puts it plainly: the larger the second number, the smaller the scale. 1/100,000 is a smaller fraction than 1/24,000, so a 1:100,000 map is a smaller-scale map. It shows a bigger area with less detail. A 1:24,000 map is large-scale: USGS describes maps at that scale as showing buildings, campgrounds, footbridges and fence lines.

A 1:100 plan shrinks area 10,000 times

A scale applies to lengths. Areas shrink by the scale squared. On a 1:100 plan, a square 1 cm by 1 cm stands for 100 cm by 100 cm, which is 10,000 square centimetres, or 1 square metre. So a room that covers 12 square centimetres on the plan covers 12 square metres in real life, and the area ratio is 1 : 10,000, not 1 : 100.