1. How Thales Calculated a Pyramid's Height from Its Shadow
It is said that the ancient Greek scholar Thales measured the height of a pyramid using its shadow [1]. He reportedly measured a massive building, which was difficult to climb or lower a rope from, using only the length of the shadow cast on the ground. There are two ways this method is said to have been done. One way was to wait for the moment when his own shadow was the same length as his height, and then measure the pyramid's shadow at that time. The other way was to stand a vertical stick in the ground at the same time, compare the stick's shadow to the pyramid's shadow, and calculate the height using the similarity of two right-angled triangles [1].
Measuring with shadows works because sunlight comes from the same direction at the same time. Since the slope of the line connecting the tip of the shadow to the top of the object is the same for both the stick and the pyramid, the ratio "stick height : stick shadow = pyramid height : pyramid shadow" holds true. If the stick's shadow is twice its height, the pyramid's shadow is also twice its height. However, this is a tradition; Thales' own writings do not survive [1]. While we cannot be certain, this shows the earliest form of the idea of replacing an unreachable height with a reachable length. But what about mountains where shadows are not useful?
Note: Right-angled triangles with the same slope are similar shapes, meaning they have the same shape but different sizes.
2. How Old Chinese Methods Measured Mountains by Lining Up Trees and Peaks
The old Chinese mathematics book "The Nine Chapters on the Mathematical Art" contains methods for measuring mountain heights. The method involves moving until the tip of a tree in front of the mountain and the mountain peak appear to overlap exactly. From that position, you measure the distance to the tree and the distance to the mountain, then calculate using the similarity of right-angled triangles [2]. Your line of sight passes through both the tree tip and the peak. Therefore, the small right-angled triangle in the foreground and the large right-angled triangle including the mountain are similar. Since the height of the reachable tree can be measured with a ruler, you can roughly determine the mountain's height using the ratio of distances. This uses the same concept of "ratios" as shadows, but it works even when the sun is not shining.
However, there is not always a tree in the perfect position. The National Diet Library explains that using a tool called a "shihou-ku" (a measuring square) instead of a comparison tree allows you to measure mountain heights from anywhere [2]. By changing the marker from a natural tree to a tool, the location for measuring became flexible. So, how accurate are these measurements when using tools on high mountains? In Japan during the Edo period, someone measured Mount Fuji.
3. Why Ino Tadataka's Angle Measurement of Mount Fuji Was About 150 Meters Off
Ino Tadataka was a person who walked all over Japan to create maps. He calculated the height of Mount Fuji from distances and directions measured on the ground, along with the angle of elevation, arriving at a value of 3,928 meters (about 12,887 feet) [3]. The value currently used is about 3,776 meters (about 12,388 feet), based on a survey from 1926, and it was corrected to 3,775.51 meters in 2014 [3]. The difference is about 150 meters, which is about 4% of the current value (calculated by dividing 152 by 3,776). The concept is the same right-angled triangle used in the tree and mountain method. If you know the angle of elevation and the horizontal distance in front of the mountain, you can calculate the height side. However, the reason for the approximately 150-meter error is not written in the referenced materials, so no reason is decided here. What is clear is that with only walked distances and viewing angles, people in the Edo period got within 4% of the current value for Mount Fuji's height. So, how is the current value determined?
4. How Leveling Surveys Find Elevation by Adding Up Small Height Differences
Current elevations in Japan are determined by leveling surveys. According to the Geospatial Information Authority of Japan, you place leveling staffs (long rulers) at two points, set a level instrument horizontally in the middle, and read the two scales. The difference between the readings is the height difference between the two points [4]. This is repeated for each section. Starting from a benchmark with a known elevation, you add up the height differences to get the elevation of a new point [4]. Think of it like stairs: since the height of each step can be checked with a ruler, counting the steps and adding them gives you the height of the upper floor. Unlike real stairs, however, the height difference for each section is found by the difference in scale readings. The starting point for this addition is the Japan Vertical Datum in Nagatacho, Chiyoda-ku, Tokyo. It was established in 1891, and its current elevation is 24.3900 meters [5]. Elevations in Japan are defined as the height above the mean sea level of Tokyo Bay [5]. Adding up small differences takes effort, but it allows progress while checking each step. How do we find the height of distant places that people cannot walk to? Today, we borrow the power of satellites.
5. How Satellite Positioning Gives Elevation by Subtracting Sea Level
GNSS does not show elevation directly. GNSS provides the height above a smooth ellipsoid model of the Earth, known as ellipsoidal height. To get elevation, you subtract the geoid height using a geoid model [4]. The geoid is described as a surface that extends the mean sea level under the land. The height from the satellite and the height above sea level, which is the standard for elevation in Japan, are numbers based on different surfaces, so they are connected by subtraction. The Geospatial Information Authority of Japan explains that from April 1, 2025, the elevation of control points was revised to values based on satellite positioning [4]. The amount that elevation drops varies by location; in estimates, it is said to be about 40 cm in the eastern part of Hokkaido. Instead of walking and adding, we find height using satellite positions and subtraction. This is also an extension of the idea of replacing an unmeasurable height with "measurable quantities." How finely are these "measurable quantities" used in the towers we build today?
6. How Tokyo Skytree Was Built with Less Than 2 cm of Error
Tokyo Skytree reached a height of 634 meters (about 2,080 feet) on March 18, 2011. The construction company, Obayashi Corporation, explains that the steel structure was assembled with an error of less than 2 cm relative to the 634 meters [8]. As a ratio, 2 divided by 63,400 is about 0.003%, or about one in 30,000. Supporting this precision was the 3D optical surveying instrument (total station) that measures the coordinates of two points on a column. The measured values were calculated on a tablet to produce correction values for twisting or tilting [8]. The process involved building up high, measuring, calculating, and correcting. Here, surveying is not just a task to know the height once, but a tool used continuously while building. So, how was Everest's elevation determined with these tools?
7. Why Mount Everest's 8,848.86 Meters Is the Height of the Snow
On December 8, 2020, China and Nepal jointly announced the elevation of Mount Everest as 8,848.86 meters. This is 0.86 meters higher than the previously recognized 8,848 meters [6]. The survey used leveling, gravity, and GPS data [6]. This means the "accumulation of small differences" and "satellites" seen so far were used together on the world's highest mountain. However, this value is the height of the summit with snow accumulated on it, known as the snow surface height. A 2023 study states that the height of the rock beneath the snow has not been determined because the thickness of the snow is unknown [7]. The answer changes depending on whether the measured surface is "snow" or "rock." This article does not determine what caused the 0.86-meter difference. Height has meaning only when accompanied by "which surface was measured."
8. How to Measure Nearby Trees or Poles Using Their Shadows
Thales' method can be reproduced near your home. On a sunny day, stand on flat ground and look at your shadow. Wait for the time when your shadow's length is the same as your height. Do not look directly at the sun. At that time, measure the shadow length of a nearby tree or utility pole using your stride or a tape measure. If the shadow and height are the same, the shadow extends at nearly the same ratio, so the shadow length is a rough guide to the object's height. Since the tip of the shadow is blurry, the value is an estimate. If you want to investigate more, the National Diet Library's "History of Mathematics" Chapter 4 (Japanese mathematics) explains how to measure mountain heights and describes the "shihou-ku" [2]. The Geospatial Information Authority's page on leveling surveys explains how to use leveling staffs and level instruments [4]. Even for the same mountain elevation, looking at which surface the height is measured from changes how the number appears.