What is Shrunk Scale in Surveying?

A shrunk scale or a Reduced scale is a correction factor that is used in surveying to account for the shrinkage of survey materials like paper or plastic on which plans and maps were drawn. These materials can shrink due to changes in temperature, humidity, and other environmental factors. 

The shrinkage of maps leads to distortions in the measurements and scales of the maps leading to the inaccuracy of the representations of the surveyed area. Hence, to determine the extent of shrinkage caused and to derive the original scale, we need to calculate the shrunk scale of a map.

In this article, we will discuss in detail the formula to calculate the shrunk scale, shrinkage factor, shrinkage ratio, and a few workout problems. 

Understanding Original Scale and Shrunk Scale

When a map or plan is created, it is drawn using an original scale, which accurately represents the real-world dimensions on a smaller scale. However, over time, the material on which the map or plan is drawn, such as paper or plastic, may shrink due to environmental factors like temperature and humidity. Let's learn one by one.

Original Scale

Let’s say, we have to plot a proper rectangular area on a map with dimensions of 50 m x 80 m. The area is marked as ABCD, where AB = 50 m and BC = 80 m. We plan to represent this area on the map with a scale where 1 centimetre on the map equals 10 meters in reality.

To calculate the Representative Fraction (R.F), we proceed as follows: 1 centimetre on the map equals 10 meters in reality. Thus, R.F = 1 cm / 10 m. When simplifying, we convert both the numerator and the denominator to the same unit. Therefore, 10 meters becomes 1000 cm. So, R.F = (Unit on the Map) / (Unit on the Field) = 1 / 1000.

Therefore, our scale is 1:1000. 

This scale that we have determined to plot the map is called the original scale.

Now, how can we represent the area on the map?

The length of side AB is 50 metres, which equals 5000 cm. On the map, we will plot AB with a measure of AB x R.F = 5000 cm x (1/1000) = 5 cm. Similarly, side BC, which is 80 metres, will be represented as 8 cm on the map.

Thus, on the map, we take 5 cm and 8 cm as the sides of the rectangle, which corresponds to 50 m and 80 m in reality, as shown on the map. In the rightmost corner, we mention the scale as 1:1000. (Fig.3).

Fig.3. A Site Plan Drawn Using a Scale 1: 1000

As we are representing a larger dimension from the field on the map using a reduced dimension, this is called a reducing scale.

Read More On : Reducing Scale, Full Scale and Enlarging Scale

To understand this better, if we have a map with no specifications given for any length, we measure the length AB using a ruler. This length is the length on the map. Using the drawing scale, i.e., the R.F., we convert it into the original length on the field.

For instance, here, when we measure AB, we get 5 cm. As per the scale, 1: 1000, 1cm on the map represents 1000 cm on the field. So how much is 5cm on the map?
Multiplying it by 1000 (1:1000) gives us 5000 cm in reality, which equals 50 m on the field.

Shrunk Scale

Now that we have understood the original scale of a map or plan, let’s study what is a shrunk scale. 
With time and atmospheric conditions, the map or plan material can shrink, which would result in changes in the measurements. If specifications of measurements are not given on the map, we measure using a scale and convert it as per the drawing scale to obtain the actual measurement on the site.
For instance, if the plot map ABCD after years shrunk and when measured, the side AB using a scale, a value of 4 cm was obtained. The scale is 1:1000. This means, that on the field AB is 40m. So we need to find the new shrunk scale so that we can apply correction. 

In the above equation, the ratio of shrunk length to the original length is called as the shrinkage factor. Hence, we get

In this example, the original length of 5 cm on the map shrunk to 4 cm. Hence, the shrunk length is 4 cm. The original scale is 1/1000. Then based on the equation;

Shrunk scale or Reduced Scale = (1/1000) x (4/5) =  1/ (5000/4) = 1/1250 = 1:1250;  

Then if only shrunk length is given, the original length = shrunk length x 1250 = 5000 cm = 50m;

Calculation of Shrunk Scale?

In order to calculate the shrunk scale, we need to determine the shrinkage factor. 

1. Shrinkage Factor (S.F)

Shrinkage Factor = Shrunk Length or Measured length / Original Length or Correct Length

(EQ.1)

Shrunk length or Measured Length is measured from the map, as the distance between the two points as per the give ratio of scale.

The original length or Correct length is taken from the map - actual distance between the two points labelled on the map. 

2. Shrunk Scale/ Reduced Scale

Given the shrinkage factor, we can determine the shrunk scale by the formula:

Shrunk Scale = Original Scale x Shrinkage Factor (S.F) 

(EQ.2)

3. Corrected Length ,Corrected Area and Corrected Volume on Map

If the map is shrunken, and we need to find out the original length or area of the area enclosed by sides on the map, we can use the following formula:

Corrected Length =  Shrunk or Measured Length / S.F

(EQ.3)

Corrected Area = Correct length x Correct Breadth

After applying EQ.3 in the above equation, we get:

Corrected Area on Map = (Measured Length/S.F ) x (Measured Breadth /S.F)

Corrected Area on Map = Measured Area/(S.F)²

^(EQ.4)

Corrected Volume on Map = Measured Volume / (S.F)³

^(EQ.5)

4. Corrected Length , Area and Volume on Field

Corrected Length on the field = Corrected length on map /( Original Scale)

[ We know that original scale R.F = Length on Map/ Length on Field]

Based on EQ.3,

Corrected Length on the field    = (Measured Length/S.F) x  (1/ Original Scale) 

                                                            = Measured Length / (S.F x Original Scale)

From EQ.2, we get

Corrected Length on the Field = Measured Length / R.F of Shrunk Scale

^(EQ.6)

Similarly,

Corrected Area on the Field = Measure Area/ (R.F of Shrunk Scale)2

^(EQ.7)

Read More On : Civil Engineering Drawings