The basic principle that was used earlier is very simple, and uses only trigonometry which most of us are familiar with, or at least can recall. There are three sides and three angles in any triangle. If we know any three of these quantities, provided one of them is a side, all the others can be calculated. In a right-angled triangle, one of the angles is already known, so if we know any other angle and one of the sides, the others can be found out. This principle can be applied for measuring the height of any object that does not offer the convenience of dropping a measuring tape from top to bottom, or if you can’t climb to the top to use sophisticated instruments.
Measurement with angles: When direct measurement of height is not possible, a surveyor resorts to trigonometry
Let’s say, we have to measure the height of a pole, or a building. We can mark any arbitrary point on the ground some distance from the building. This can be our point of observation. We now need two things — the distance of the building from the point of observation, and the angle of elevation that the top of the building makes with the point of observation on the ground. The distance is not difficult to get. The angle of elevation is the angle that an imaginary line would make if it was joining the point of observation on the ground to the top of the building. There are simple instruments with the help of which this angle can be measured.
So, if the distance from the point of observation to the building is d and the angle of elevation is E, then the height of the building would be d × tan(E).
The ExpertsLt Gen Girish Kumar is the Surveyor General of India, and Nitin Joshi is Deputy Surveyor General, Survey of India. The responsibility of the Survey of India is to prepare authoritative maps, and its work involves carrying out extensive land surveys and mapping topographical features. Starting in 1952, the Survey of India undertook an exercise to measure the height of Mount Everest (then known as Peak XV). That exercise measures the height at 8,848 m (29,028 feet), which remained the globally accepted standard, until now.
Can it be that simple for measuring a mountain?
The principle is the same, and ultimately, we use the same method, but there are a few complications. The main problem is that though you know the top, the base of the mountain is not known. The question is from which surface you are measuring the height. Generally, for practical purposes the heights are measured above mean sea level (MSL). Moreover, we need to find the distance to the mountain. It seems easy today, but there were no GPS or satellite images in the 1950s. So, how to find the distance of a mountain where you cannot physically go? Till that time nobody had even climbed the Mount Everest.
We can get around this problem by measuring the angles of elevation from two different points of observation in the same line of view. The distances between these different points of observations can be measured. We will now be dealing with two different triangles, but with a common arm, and two different angles of elevation. Again, by following simple rules of high-school trigonometry, the height of the mountain can be calculated, fairly precisely. In fact, this is how we used to do it before the advent of GPS, satellites and other modern techniques.
How accurate is this?
For small hills and mountains, whose top can be observed from relatively close distances, this can give quite precise measurements. But for Mount Everest and other high mountains, there are some other complications.
These again arise from the fact that we do not know where the base of the mountain is. In other words, where exactly does the mountain meet flat ground surface. Or, whether the point of observation and the base of the mountain at the same horizontal level.
The Earth’s surface is not uniformly even at every place. Because of this, we measure heights from mean sea level. This is done through a painstaking process called high-precision levelling. Starting from the coastline, we calculate step by step the difference in height, using special instruments. This is how we know the height of any city from mean sea level.
But there is one additional problem to be contended with — gravity. Gravity is different at different places. What that means that even sea level cannot be considered to be uniform at all places. In the case of Mount Everest, for example, the concentration of such a huge mass would mean that the sea level would get pulled upwards due to gravity. So, the local gravity is also measured to calculate the local sea level. Nowadays sophisticated portable gravitometers are available that can be carried even to mountain peaks.
But the levelling cannot be extended to high peaks. So we have to fall back on the same triangulation technique to measure the heights. But there is another problem. The density of air reduces as we go higher. This variation in air density causes the bending of light rays, a phenomenon known as refraction. Due to the difference in heights of the observation point and the mountain peak, refraction results in an error in measuring the vertical angle. This needs to be corrected. Estimating the refraction correction is a challenge in itself. 📣 Follow Express Explained on Telegram
In this May 27, 2020 photo, members of a Chinese surveying team head for the summit of Mount Everest, also known locally as Mt. Qomolangma. (AP/PTI Photo)
Doesn’t technology offer easier solutions?
These days GPS is widely used to determine coordinates and heights, even of mountains. But, GPS gives precise coordinates of the top of a mountain relative to an ellipsoid which is an imaginary surface mathematically modelled to represent Earth. This surface differs from mean sea level. Similarly, overhead flying planes equipped with laser beams (LiDAR) can also be used to get the coordinates.
But these methods, including GPS, do not take gravity into consideration. So, the information obtained through GPS or laser beams is then fed into another model that account for gravity to make the calculation complete.
Considering that during 1952-1954, when neither GPS and satellite techniques were available nor the sophisticated gravimeters, the task of determining the height of Mount Everest was not easy.
Nepal and China have said they have measured Mount Everest to be 86 cm higher than the 8,848 m that it was known to be. What would that mean?
The 8,848-metre (or 29,028-foot) measurement was done by the Survey of India in 1954 and it has been globally accepted since then. The measurement was carried out in the days when there was no GPS or other modern sophisticated instruments. This shows how accurate they were even during that time.
In recent years, several attempts have been made to re-measure Everest, and some of them have been produced results that vary from the accepted height by a few feet. But these have been explained in terms of geological processes that might be altering the height of Everest. The accuracy of the 1954 result has never been questioned.
Most scientists now believe that the height of Mount Everest is increasing at a very slow rate. This is because of the northward movement of the Indian tectonic plate that is pushing the surface up. It is this very movement that created the great Himalayan mountains in the first place. It is this same process that makes this region prone to earthquakes. A big earthquake, like the one that happened in Nepal in 2015, can alter the heights of mountains. Such events have happened in the past. In fact, it was this earthquake that had prompted the decision to re-measure Everest to see whether there had been any impact.
A 86-cm rise would not be surprising. It is very possible that the height has increased in all these years. But, at the same time, 86 cm in a height of 8,848 metres is a very small length. The detailed results of the Nepali and Chinese efforts at measuring Everest are still to be published in a journal. The real significance of this measurement would become evident only after that.
A bird flies with Mount Everest seen in the background from Namche Bajar, Solukhumbu district, Nepal. (AP/PTI Photo) 
Measurement with angles: When direct measurement of height is not possible, a surveyor resorts to trigonometry
In this May 27, 2020 photo, members of a Chinese surveying team head for the summit of Mount Everest, also known locally as Mt. Qomolangma. (AP/PTI Photo)
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