A mountain or anything for that matter could be thousands of kilometres in land. How do they do it?
In: Engineering
There is something called a theodolite which in a basic form was used by the Romans these have become far more advanced in the last 100 years.
They employed calculators. No, not the kind with a screen and buttons to push, actual human beings with pencils and paper, whose job it was to take the measurements from surveyors, do the mathematical calculations (trigonometry, etc.) and come up with the result.
Fun fact, the original pencil and paper calculations for the height of Mount Everest were found to only be off by about 2-3 feet, when compared to satellite images used in modern times to get an accurate number.
You can use angles and trigonometry to measure things with line of sight
You can fill a tube with water, add all the twists and turns you want, and the two ends of the pipe will have water at the same level. We even still use this method today.
Mostly, we use surveying equipment with lasers. It’s all the same principles, but just more precise.
Trigonometry. After not using trig for 30 years, I used it to find out the mast at my new house was 28 feet tall. 🙂
You could also calculate altitude using a barometer. This process was invented in ~1775, and you use an equation to convert your barometers reading to altitude. Generally these calcs were printed in books so you did a quick cross reference to get your answer.
This is the basis of the pressure alitimeter that is used in aircraft and by sky divers to this day.
You might like the story of how Radhanath Sikdar, working for British Surveyor General George Everest, during the [Great Trigonometrical Survey of India](https://en.wikipedia.org/wiki/Great_Trigonometrical_Survey), used a triangulation system involving [theodolites](https://en.wikipedia.org/wiki/Theodolite) and [surveying chains](https://en.wikipedia.org/wiki/Gunter%27s_chain), with very careful correction for factors such as temperature on the chains, to near perfectly estimate the height above sea level of Mt. Everest, though it is very far from the sea.
Sikdar came up with a figure of exactly 29,000 feet. So they added two feet, thinking no one would ever believe the even number.
NPR, on [Everest measurement, then and now](https://www.npr.org/2020/11/24/938736955/how-tall-is-mount-everest-hint-its-changing), with some of the story
Trigonometry. Take a reference point like sea level. Measure triangle angles very accurately and use math to calculate the distances. Surveyors would use rods and chains to measure known distances across ground to set baselines. Then you measure angles from those reference points to get the numbers.
Today, the measurements are largely electronic with total stations taking the distance and angle measurements with lasers. Also GPS measurements can be done with survey beacons providing a fixed reference point to compensate for GPS inaccuracy.
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