Measurement and construction (determination) of distances on a topographic map
Map scale
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 its corresponding length on the ground. It is expressed as the ratio of two numbers. For example, a scale of 1:50,000 means that all terrain lines are shown on the map with a reduction of 50,000 times, i.e. 1 cm on the map corresponds to 50,000 cm (or 500 m) in the terrain.
The scale is indicated below the bottom of the map frame in digital terms (numerical scale) and in a straight line (linear scale), on the segments of which the corresponding distances on the ground are signed (Fig. 1). It also indicates the magnitude of the scale  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 zeros on the right side of the relationship, then the remaining number will show how many meters in the area correspond to 1 cm on the map, i.e., the magnitude of the scale.
When comparing multiple scales, the larger will be the one with a smaller number on the right side of the relationship. Suppose that maps of the scales of 1: 25000, 1: 50000 and 1: 100000 are available for the same area. Of these, a scale of 1: 25,000 will be the largest, and a scale of 1: 100,000 will be the smallest.
The larger the scale of the map, the more detailed the terrain is depicted on it. With a decrease in the scale of the map, the amount of terrain features applied to it also decreases.
The detail of the 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: 10000, 1: 25000, 1: 50000, 1: 100000, 1: 200000, 1: 500000 and 1: 1,000,000.
The cards used by the troops are divided into largescale, mediumscale, and smallscale.
Measurement on a map of straight and curving lines  

Map scale 
Card name 
Card classification 

in scale 
for 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 five thousandth 

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

1: 100 000 (in 1 cm 1 km) 
hundred thousandth 
mediumsized 

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: 1 000 000 (in 1 cm 10 km) 
millionth 
Measurement on a map of straight and curving lines
When measuring straight lines, compass needles are set to end points, then, without changing the compass solution, the distance is taken on 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 in the usual order on a linear or transverse scale.
It is convenient to measure broken lines by successively building up the compass solution in straight sections. The distance corresponding to the compass solution is determined by the order described above.
Curve distances are measured in compass steps. The compass step length depends on the degree of tortuosity of the line, but, as a rule, should not exceed (for accurate measurements) 1 cm. To exclude errors due to deformation of the paper (card), the compass step length is preliminarily checked along the kilometer grid line. To measure distances on the map using a curvimeter, you must first (by rotating the wheel) set the arrow to zero (initial) division, then roll the wheel with uniform pressure from the source to the end point. At the same time, you should pay attention to the fact that when moving the curvimeter, the readings of the track count increase, but not decrease; otherwise, the curvimeter must be rotated 180 °. If the scale of the curvimeter is signed in kilometers, the distance obtained is read directly from the scale.
If the scale of the curvimeter is signed in kilometers, the distance obtained is read directly from the scale. If the scale divisions are given in centimeters of the wheel path on the map, then the obtained number of divisions must be multiplied by the division price. In order to avoid errors, it is recommended that the division price be determined by control measurement along the kilometer grid line.
When measuring straight lines, compass needles are set at the end points, then, without changing the compass solution, the distance is measured on a linear or transverse scale:
 Determining distances on a linear scale using a compass In the case when the compass solution 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 is determined in the usual order on a linear or transverse scale. It is convenient to measure broken lines by successively building up the compass solution in straight sections.
 Measurement of distances by way of building up the compass solution The distance corresponding to the compass solution is determined by the procedure described above. Curve distances are measured in compass steps. The compass step length depends on the degree of tortuosity of the line, but, as a rule, should not exceed (for accurate measurements) 1 cm. To exclude errors due to deformation of the paper (card), the compass step length is preliminarily checked along the kilometer grid line.
To determine the distance between points of the terrain (objects, objects) from the map, using the numerical scale, you need to measure the distance between these points in centimeters on the map and multiply the resulting number by the magnitude of the scale.
Example, on a map with a scale of 1: 25000, we measure with a ruler the distance between the bridge and the windmill (Fig. 2); it is 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 on the map the distance between points of the area 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 sufficient, the solution of which is equal to the distance between the given points on the map, applied to a linear scale and take a count in meters or kilometers. In fig. 3, the measured distance is 1070 m.
Fig. 3. Measurement on a distance map with a compass meter on a linear scale. 
Fig. 4. Measurement on a map of distances with a compass meter along winding lines. 
Large distances between points in straight lines are usually measured using a long ruler or compass gauge.
In the first case, to determine the distance on the map using the ruler, use a numerical scale (see. Fig. 2).
In the second case, the “step” solution of the compass meter is set so that it corresponds to an integer number of kilometers, and an integer number of “steps” is set aside on the segment measured on the map. The distance that does not fit into the integer number of "steps" of the compass gauge is determined using a linear scale and added to the obtained number of kilometers.
In the same way, distances are measured along winding lines (Fig. 4). In this case, the "step" of the compass meter should be taken 0.5 or 1 cm, depending on the length and degree of tortuosity of the measured line.
Fig. 5. Distance measurements with a curvimeter
To determine the length of the route on the map, a special device called a curvimeter (Fig. 5) is used, which is especially convenient for measuring winding and long lines.
The device has a wheel, which is connected by a gear system to the arrow.
When measuring a distance with a curvimeter, you need to set its arrow by division 99. Keeping the curvimeter in an upright position, guide it along the measured line, without taking it off the map along the route so that the scale readings increase. When 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).
The accuracy of measuring distances on the map. Correction for the distance for the slope and curvature of the 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 selected measurement method, terrain and other factors. You can most accurately determine the distance on the map in a straight line.
When measuring distances using a compass gauge or a ruler with millimeter divisions, the average value of the measurement error on flat terrain usually does not exceed 0.71 mm on a map scale, which is for a map of scale 1: 25000  17.525 m, scale 1: 50,000  3550 m, scale 1: 100000  70100 m.
In mountainous areas with a large slope slope, errors will be greater. This is due to the fact that when shooting the terrain on the map, it is not the length of the lines on the Earth’s surface that is plotted, but the length of the projections of these lines onto the plane.
For example, with a slope steepness of 20 ° (Fig. 6) and a distance of 2120 m on the terrain, its projection onto a plane (distance on the map) is 2000 m, i.e., 120 m less. It is estimated that at an inclination angle (slope steepness) of 20 °, the result of measuring the distance on the map should be increased by 6% (add 100 m per 100 m), at an inclination angle of 30 °  by 15%, and at an angle of 40 °  by 23 %
Fig. 6. The projection of the length of the slope on a plane (map)
When determining the route length on a map, it should be borne in mind that the distances along the roads measured on the map with a compass or curvimeter are, in most cases, shorter than the actual distances.
This is explained not only by the presence of descents and ascents on the roads, but also by some generalization of the meanders of the roads on the maps. Therefore, the result of measuring the route length obtained from the map should be multiplied by the coefficient indicated in the table, taking into account the nature of the terrain and the scale of the map.
The simplest ways to measure area on a map  

Nature of the area 
The coefficient of increasing the length of the route, measured on a scale map 

1: 50,000 
1: 100000 
1: 200000 

Mountain (strongly intersected) 
1.15 
1.20 
1.25 
Hilly (midcut) 
1.05 
1.10 
1.15 
Plain (slightly intersected) 
1.00 
1.00 
1.05 
An approximate assessment of the size of the areas is performed by eye on the squares of the kilometer grid available on the map. Each square of the grid of maps of scales of 1: 10000  1: 50,000 on the ground corresponds to 1 km2, the square of the grid of maps of scale of 1: 100000  4 km2, 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).
Having imposed such a palette on the measured object on the map, it is first calculated by 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 the squares, the area of the object is obtained.
On squares of scales 1: 25000 and 1: 50,000, the areas of small sections are conveniently measured 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 cart.
Definition of distances by rectangular coordinates
The straight line distance D between points with known rectangular coordinates given in the same system can most accurately be 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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