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Surveyors measured mountain heights with simple trigonometry. From two ground stations at a known distance apart, they used a theodolite (a precision protractor) to measure the angles up to the summit. Knowing two angles and the baseline length, the height of the peak falls out of the sine rule. The ancient Greeks did this for towers and pyramids; Victorian surveyors did it for the Himalayas before GPS satellites took over the job.
One often-told story has it that when the Welsh-born Surveyor General of India, Sir George Everest, measured Mount Everest, then known as Peak XV, to be exactly 29,000 feet tall, his team added two feet to make the number look less round. The reality is a bit different: the height was actually calculated in 1852 by Radhanath Sikdar, the Chief Computer of the Great Trigonometrical Survey, working under Everest’s successor Andrew Scott Waugh. Sikdar landed on 29,000 ft and Waugh added the two extra feet before formally announcing 29,002 ft in March 1856. Everest himself had retired from the Survey in 1843 and never measured the mountain that now bears his name.
Modern technologies like GPS satellites have refined that figure several times since—a 1999 American GPS expedition put Everest at 29,035 ft, and the joint China–Nepal survey announced in December 2020 fixed the currently accepted height at 8,848.86 m (29,031.7 ft). The Survey of India’s 1852 result is still extraordinary given the constraints: Sikdar and his team had no GPS, and they worked from observation stations more than 150 miles south of the peak, because Nepal at the time refused entry to British surveyors. So how did they pull it off?
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Trigonometry
As a child, the oldest method I can remember to determine length was to scale it with my hand. A single unit of measurement would be the distance between my thumb and the pinky when the hand was moderately stretched. To measure, say, a table, I would place my stretched hand upon it. Next, the pinky would leap forward, allowing the thumb to take its place, and the length would then measure two units. The scampering would continue until the entire length was accounted for.
Eventually, hands were replaced by rulers, but the methodology remained the same – stack one beside another until the entire length is covered. No one will deny that measuring how tall Mount Everest is with a ruler or your hand is possible, but I’m sure that everyone would agree that the process would be a bit time-consuming and cumbersome.
Yet, the method that geographers rely on isn’t far from using rulers. In fact, Sir Everest and his team implemented high-school geometry to measure the elevation of Mount Everest. Yes, that’s right, their tools were just fancier, more sophisticated sets of rulers and protractors. Trigonometry had been used by the Greeks to measure tall structures and by Victorian surveyors to measure the tallest mountains before we moved onto satellites. However, even satellites measure elevation by essentially implementing the same principle – drawing triangles.
Triangles
Geographers measure elevation by drawing numerous triangles. Amongst the three sides, one is the altitude of the mountain whose elevation must be measured. The base of a triangle is drawn between the mountain’s feet and a point, let’s say A, that is situated at a known distance from the mountain’s feet. The third side can be formed by simply connecting point A and the summit.
While forming the horizontal base, geographers must ensure that it is completely level to achieve accurate results. The discernment of any irregularity on Earth’s craggy surface is achieved with the help of highly delicate instruments. Next, they must measure all three angles formed within the triangle. This is achieved by using an advanced protractor, known as a theodolite. Measuring even two angles is sufficient, as the third angle can be calculated by subtracting the sum of the two known angles from 180, as the sum of all three angles bounded by a triangle is equal to 180º.
Now, behold the magic of simple trig — the knowledge of two angles and the length of one side can reveal the altitude of the mountain. Even the Greeks measured elevation by “comparing the ratios of two sides of a triangle”, which, if you’ve learned basic trig, is essentially performing the same operation.
For instance, consider a very simple example where the angle formed at point A is 60º, and we only know the distance between point A and the mountain’s base, which is, of course, the triangle’s base. For simplicity, let’s assume the triangle is a right-angled, where the base is perpendicular to the altitude. This implies that the third angle, formed at the summit is 30º (180º-[90º+60º]). Let’s also label the triangle’s sides. Starting from the altitude and going clockwise, let’s label them as X, Y and Z units.
Now, Sin (60º) represents the ratio X/Y, while Sin (30º) represents the ratio Z/Y. If we divide these ratios, we observe that the two Ys cancel out and we are only left with the ratio X/Z. The values of both Sin (60º) and Sin (30º) can be learned by simply referring to a high school math textbook. Furthermore, Z is the base of the triangle, the magnitude of whose length we already know. Multiply Z with the ratios of Sines and we have the altitude – X — of the mountain.
Sikdar and the Survey of India drew several such triangles, all emanating from different observation stations on the Indian plains, as the measurement from any one triangle could not be trusted. The team averaged the altitudes derived from all the triangles, which gave a final figure of exactly 29,000 feet—a suspiciously round number that Andrew Waugh nudged up to 29,002 ft in the official announcement so it would look like a measurement and not a guess.
In 1999, an American GPS expedition led by Bradford Washburn put Everest at 29,035 ft (8,850 m). Two decades later, in December 2020, China and Nepal jointly fixed the currently accepted height at 8,848.86 m (29,031.7 ft). Either way, the 1852 figure is remarkable—Sikdar’s answer was within about 30 feet of the modern reading. Only two angles and one side, and that was it.
