Measurement and construction (determination) of distances on a topographic map
Map Scales
Fig. 1. Design of numerical and linear scales on topographic maps and city plans
The scale of the map shows how many times the length of the line on the map is less than the corresponding length on the ground. It is expressed as a ratio of two numbers. For example, a scale of 1:50 000 means that all terrain lines are depicted on the map with a decrease of 50,000 times, i.e. 1 cm on the map corresponds to 50,000 cm (or 500 m) on the terrain.
The scale is indicated under the bottom side of the map frame in numerical terms (numerical scale) and as a straight line (linear scale), on the segments of which the corresponding distances on the ground are signed (Fig. 1). It also indicates the scale value  the distance in meters (or kilometers) on the ground, corresponding to one centimeter on the map.
It is useful to remember the rule: if you cross out the last two zeroes on the right side of the relationship, the remaining number will show how many meters in the terrain corresponds to 1 cm on the map, i.e. the magnitude of the scale.
When comparing several scales, the one with the number on the right side of the relation is smaller. Suppose that for the same piece of terrain there are maps of scales 1: 25000, 1: 50,000 and 1: 100,000. Of these, the scale of 1: 25,000 will be the largest, and the scale of 1: 100,000 will be the smallest.
The larger the scale of the map, the more detail it shows the terrain. With a decrease in the scale of the map, the number of terrain details applied to it decreases.
The detail of a terrain image on topographic maps depends on its nature: the less details the terrain contains, the more fully they are displayed on maps of smaller scales.
In our country and many other countries, the main scales of topographic maps are: 1: 10,000, 1: 25,000, 1: 50,000, 1: 100,000, 1: 200,000, 1: 500,000 and 1: 1,000,000.
The maps used in the troops are divided into largescale, mediumscale and smallscale.
Measurement on the map of straight and winding lines  

Map scale 
Card Name 
Card classification 

by scale 
according to the main purpose 

1:10 000 (in 1 cm 100 m) 
ten thousandth 
large scale 
tactical 
1:25 000 (in 1 cm 250 m) 
twenty thousandth 

1:50 000 (in 1 cm 500 m) 
five thousandth 

1: 100,000 (in 1 cm 1 km) 
one hundred thousandth 
medium scale 

1: 200,000 (in 1 cm 2 km) 
two hundred thousandth 
operational 

1: 500,000 (in 1 cm 5 km) 
five hundred thousandth 
small scale 

1: 1 000000 (in 1 cm 10 km) 
millionth 
Measurement on the map of straight and winding lines
When measuring straight lines, the compass needle is placed on the end points, then, without changing the compass solution, the distance is taken along a linear or transverse scale. In the case when the compass solution exceeds the length of the linear or transverse scale, a certain integer number of kilometers is determined by the squares of the kilometer grid, and the remainder is determined by the usual order of the linear or transverse scale.
It is convenient to measure broken lines by successively increasing the compass solution with straightline segments. The distance corresponding to the compass solution is determined by the order outlined above.
Measurement of distances along curves is made by compass pitch. The length of the compass pitch depends on the degree of tortuosity of the line, but, as a rule, should not exceed (for accurate measurements) 1 cm. To eliminate errors due to deformation of the paper (map), the compass pitch length is preliminarily checked along the kilometer grid line. To measure distances on the map using an odometer, turn the dial to the zero (initial) division, then roll the wheel with uniform pressure from the starting point to the end point. At the same time, it is necessary to pay attention to the fact that when advancing a curvimeter, the readings of the track account increased, and not decrease; otherwise, the odometer must be rotated 180 °. If the scale of the odometer is signed in kilometers, the resulting distance is read directly from the scale.
If the scale of the odometer is signed in kilometers, the resulting distance is read directly from the scale. If the scale divisions are given in centimeters of the wheel path on the map, then the resulting number of divisions must be multiplied by the division price. In order to avoid an error, the price of division is recommended to be determined by reference measurement along the kilometer grid line.
When measuring straight lines, the compass needle is placed on the end points, then, without changing the compass solution, the distance is taken along a linear or transverse scale:
 Determination of distances on a linear scale using a compass In the case when the solution of a compass exceeds the length of a linear or transverse scale, a certain integer number of kilometers is determined by the squares of the kilometer grid, and the remainder by the usual order of the linear or transverse scale. It is convenient to measure broken lines by successively increasing the compass solution with straightline segments.
 Measurement of distances by the method of building up a compass solution The distance corresponding to a compass solution is determined by the order outlined above. Measurement of distances along curves is made by compass pitch. The length of the compass pitch depends on the degree of tortuosity of the line, but, as a rule, should not exceed (for accurate measurements) 1 cm. To eliminate errors due to deformation of the paper (map), the compass pitch length is preliminarily checked along the kilometer grid line.
In order to determine the distance between points of a terrain (objects, objects) using a numerical scale, you must measure the distance between centimeters between these points and multiply the resulting number by the magnitude of the scale.
For example, on a 1: 25000 scale map, measure the distance between the bridge and the windmill with a ruler (Fig. 2); it is equal to 7.3 cm, multiply 250 m by 7.3 and get the desired distance; it is equal to 1825 meters (250x7.3 = 1825).
Fig. 2. Determine the distance between the points of the terrain using a ruler.
A small distance between two points in a straight line is easier to determine using a linear scale (Fig. 3). To do this, a compass meter is enough, the solution of which is equal to the distance between the given points on the map, attach to the linear scale and take a reading in meters or kilometers. In fig. 3 The measured distance is 1070 m.
Fig. 3. Measurement on the map by a linear compass gauge. 
Fig. 4. Measurement on the map by the caliper gauge along twisting lines. 
Large distances between points along straight lines are usually measured using a long ruler or a caliper gauge.
In the first case, to determine the distance on the map using a ruler using the numerical scale (see. Fig. 2).
In the second case, the “step” of the compass gauge is set so that it corresponds to an integer number of kilometers, and an integer number of “steps” is deposited on the segment measured on the map. The distance that does not fit into an integer number of “steps” of a compass meter is determined using a linear scale and added to the resulting number of kilometers.
In the same way they measure distances along winding lines (Fig. 4). In this case, the “pitch” of the measuring caliper should be taken 0.5 or 1 cm, depending on the length and degree of tortuosity of the measured line.
Fig. 5. Distance measurement with a odometer
To determine the length of the route on the map, a special device, called an odometer (Fig. 5), is used, which is especially convenient for measuring tortuous and long lines.
The device has a wheel that is connected by a gear system with an arrow.
When measuring the distance with the odometer, you need to set its arrow on division 99. Holding the odometer in a vertical position, guide it along the measured line, without tearing it from the map along the route so that the scale readings increase. Reaching the end point, count the measured distance and multiply it by the denominator of the numerical scale. (In this example, 34x25000 = 850000, or 8500 m).
Accuracy of measuring distances on the map. Corrections for distance for slope and tortuosity of lines
The accuracy of determining distances on the map depends on the scale of the map, the nature of the measured lines (straight, winding), the chosen method of measurement, the terrain, and other factors. Most accurately determine the distance on the map can be in a straight line.
When measuring distances using a compass gauge or a ruler with millimeter divisions, the average measurement error on flat terrain usually does not exceed 0.71 mm on a map scale, which is 17.525 m for a 1: 25000 map; the scale is 1: 50000  3550 m, the scale is 1: 100000  70100 m.
In mountainous areas with a large steepness of slopes errors will be more. This is explained by the fact that when taking pictures of a terrain, not the length of lines on the surface of the Earth is plotted on the map, but the length of the projections of these lines onto the plane.
For example, With a slope of 20 ° (Fig. 6) and a distance of 2120 m on the ground, its projection onto a plane (distance on the map) is 2000 m, i.e., 120 m less. It is estimated that with a slope angle (slope steepness) of 20 °, the result of measuring the distance on the map should be increased by 6% (6 meters added by 100 meters), by 15% for slopes of 30 °, and by 23 degrees %
Fig. 6. The projection of the length of the slope on the plane (map)
When determining the length of the route on the map, it should be borne in mind that the distances along the roads, measured on the map with a caliper or an odometer, in most cases are shorter than the actual distances.
This is explained not only by the descents and ascents on the roads, but also by some generalization of the convolutions of the roads on the maps. Therefore, the result of measuring the length of a route derived from the map should be multiplied by a factor indicated in the table, taking into account the nature of the terrain and the scale of the map.
The simplest ways to measure areas on the map  

Terrain 
The ratio of the increase in the length of the route, measured on a map of scale 

1: 50,000 
1: 100,000 
1: 200,000 

Mountain (rugged) 
1.15 
1.20 
1.25 
Hilly (midland) 
1.05 
1.10 
1.15 
Plain (slightly intersected) 
1.00 
1.00 
1.05 
Approximate estimate of the size of the area produced by eye squares kilometer grid, available on the map. Each square of the grid of maps of scales 1: 10000  1: 50000 on the ground corresponds to 1 km2, to the square of the grid of maps of scale 1: 100,000  4 km2, to the square of the grid of maps of scale 1: 200000  16 km2.
More precisely, the areas are measured with a palette, which is a sheet of transparent plastic coated with a grid of squares with a side of 10 mm (depending on the scale of the map and the required measurement accuracy).
Putting such a palette on the measured object on the map, firstly calculate the number of squares that fit completely inside the contour of the object, and then the number of squares intersected by the contour of the object. Each of the incomplete squares is taken as half a square. As a result of multiplying the area of one square by the sum of squares, get the area of the object.
By squares of 1: 25000 and 1: 50,000 squares in small areas, it is convenient to measure with an officer's ruler, which has special rectangular cutouts. The areas of these rectangles (in hectares) are indicated on the ruler for each scale of the scale.
Determination of distances by rectangular coordinates
The distance D along a straight line between points with known rectangular coordinates, given in one system, can be most accurately determined by the formula:
D=V(Х2Х1)2+(Y2Y1)2
Where:
 X1, Y1  coordinates of the starting point;
 X2, Y2  coordinates of the end point.
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