Map Scales and Scale Drawings: Converting Map Distance to Real Distance

A map or plan is a scaled-down copy of the real world. Every measurement on it is in proportion to the real thing, so a single ratio, the scale, lets you turn centimeters on paper into kilometers on the ground or feet in a building.

Three ways to show scale

Ratio (representative fraction): 1 : 24,000 means 1 unit on the map equals 24,000 of the same units in reality. It works with any unit.
Verbal scale: "1 inch = 2,000 feet." Easy to read but tied to specific units.
Bar scale: a ruler printed on the map. It stays correct even if the map is enlarged or shrunk when printed.

The USGS 7.5-minute topographic maps used by hikers in the US are at 1 : 24,000, so 1 inch on the map is 24,000 inches, or exactly 2,000 feet.

Map distance to real distance

real distance = map distance × scale denominator

On a 1 : 50,000 map, two villages are 7.4 cm apart. Real distance = 7.4 × 50,000 = 370,000 cm = 3.7 km. Converting units at the end is where most errors happen, so remember: 100,000 cm = 1 km and 63,360 inches = 1 mile.

Using a proportion

With a verbal scale, set up a proportion. If 2 cm = 5 km and two towns are 13 cm apart:

2 cm / 5 km = 13 cm / x km

2x = 65, so x = 32.5 km. The proportion calculator solves it and shows the steps.

Real distance to map distance

Divide instead. A 12 km trail drawn on a 1 : 100,000 map is 1,200,000 cm ÷ 100,000 = 12 cm long.

Large-scale vs. small-scale maps

Confusingly, a "large-scale" map shows a small area in great detail (such as 1 : 10,000), while a "small-scale" map shows a large area with less detail (such as 1 : 1,000,000). The terms refer to the size of the fraction 1/10,000, which is larger than 1/1,000,000.

Areas on a map

Area scales by the square of the scale factor. On a 1 : 10,000 map, a rectangle 3 cm × 2 cm (6 cm²) represents 300 m × 200 m = 60,000 m², or 6 hectares. Multiplying 6 cm² by 10,000 would give the wrong answer; you need 6 × 10,000² cm². For more on area conversions see square feet and square meters.

Making a scale drawing

To draw a 14 ft × 11 ft room on graph paper at ¼ inch = 1 foot:

14 ft × 0.25 in = 3.5 inches
11 ft × 0.25 in = 2.75 inches

Architects' plans commonly use ¼ inch = 1 foot (1 : 48) for floor plans and smaller scales for site plans. Model builders use ratios such as 1 : 87 (HO trains) and 1 : 24.

Measuring curved routes

Roads and trails rarely run straight. Lay a piece of string along the route on the map, straighten it against a ruler, then apply the scale. Or measure short straight segments and add them up.

Scale drawings are an application of similar figures; see similar triangles and proportions.