The simplest way, and one that is still in use with the aid of satellites, is just an elaboration of how scouts are taught to measure the height of trees, by constructing a triangle with some known values and then filling in the blanks.
In the surveying method called triangulation, a baseline of a known length is established.
To determine the distance between any two points, they are made the vertices of a triangle.
The angle of the triangle and the length of its other two sides can be determined from observations made at the ends of the baseline.
In surveying a large area, a series of triangles is constructed upon the baseline; each has at least one side in common with adjacent triangles.
In fact, Mount Everest was named for sir George Everest, a director of the nineteenth-century Great Trigonometrical Survey of India done by this method.
A 1987 Italian expedition to confirm the height of Mount Everest made multiple observations from four ground stations on the mountain and with repeated signals from four Global Positioning System satellites.
The signals were picked up by receivers that measured the exact time the coded signal was received and computed the time difference between transmission and reception.
This time, multiplied by the speed of light, yielded the ground station’s distance from the satellite.
Four such measurements define the station’s latitude and longitude and, with further calculations, determine its elevation above sea level.
The Italians then followed regular triangulation methods to measure the height of the summit from these carefully plotted baselines.
The result: 29,108 feet above sea level, but it may be slightly higher now.
The Indian and Eurasian plates, colliding to form the Himalayas, are still pushing the chain up by half an inch or so a year.