A tower is too tall for your tape measure.
You cannot climb it. You cannot reach the top.
But you can reach its shadow.
That is enough.
Quick Answer
Put a shorter object nearby. Measure its height and its shadow.
At the same time, measure the tall object's shadow.
Because the Sun gives both objects the same light direction, the two height-and-shadow triangles have the same shape.
Their corresponding sides are proportional.
If two triangles have the same shape, a length you can measure can reveal a length you cannot reach.
The Triangle Hiding Inside a Shadow
A vertical stick and its shadow make a right triangle.
The stick is one side. The shadow is another. A ray of sunlight connects the top of the stick to the end of the shadow.
A tall tower makes the same kind of triangle.
If both shadows are measured at the same place and time, the sunlight has the same direction for both objects.
Both triangles have a right angle. They also share the same sunlight angle.
That makes them similar triangles. Similar triangles have equal corresponding angles and proportional corresponding sides.[1]
Figure 1. Same sunlight angle gives the two right triangles the same shape.
The Whole Method Is One Ratio
h = height of the small stick
s = length of the stick's shadow
H = height we want to know
S = shadow of the tall object
H / S = h / s
Solve for the unknown height:
H = h × S / s
That is the entire method.
Try One
Suppose a stick is 1.5 m tall. Its shadow is 1.0 m long.
At the same moment, a tower casts a 20 m shadow.
H = 1.5 × 20 / 1.0 = 30 m
We never touched the top of the tower.
We measured a small object, a pair of shadows, and a ratio.
A Story About Thales
A famous story says that Thales of Miletus measured the height of an Egyptian pyramid using its shadow.
But the story comes from writers who lived centuries after Thales, so we should treat the details carefully.
Different later accounts give different versions. One says he waited until a person's shadow equaled that person's height. Another describes a stick and two proportional triangles — much closer to the similar-triangle method used in this article.[2]
Later traditions credit Thales with measuring a pyramid by shadows, but we cannot know the exact method with certainty.
What Eratosthenes Added
The previous article used a shadow too. But Eratosthenes used it differently.
Here, similar triangles compare one object's height with another object's height.
Eratosthenes used a local shadow angle as a fraction of Earth's full circle.
The common habit is more important than the exact geometry:
measure something reachable → use a geometric relation → infer something unreachable
Python — Make the Ratio Reusable
def unknown_height(reference_height, reference_shadow, object_shadow):
return reference_height * object_shadow / reference_shadow
height = unknown_height(
reference_height=1.5,
reference_shadow=1.0,
object_shadow=20.0
)
print(height, "m")Predict first.
Keep the tower shadow at 20 m. Change the reference shadow from 1.0 m to 2.0 m, while the reference stick remains 1.5 m tall.
Should the calculated tower become taller or shorter? What does that say about the Sun's angle?
The Assumptions Matter
The simple formula works only when the geometry matches the model.
- Both objects are vertical.
- The ground is treated as level.
- The shadows are measured under the same sunlight direction.
- The shadow endpoints can be measured clearly.
Change those conditions and the simple ratio may need correction.
The equation is only as useful as our understanding of its assumptions.
Where This Idea Lives Today
Modern surveyors use much more precise instruments than shadow sticks.
Cameras and computers also let us measure objects we cannot easily touch.
NASA defines photogrammetry as determining the position and shape of objects from two or more images taken from different perspectives.[3]
Modern photogrammetry is not the same calculation as the shadow method. It uses camera geometry, calibration, image coordinates, and triangulation.
But the deeper strategy is familiar:
Do not touch the object. Measure a geometric projection of it, then use known relationships to recover the hidden size or position.
NASA has used photogrammetric measurements for engineering tasks such as surface-damage measurement, launch-vehicle alignment checks, and structural-motion analysis.[3]
USGS photogrammetry similarly uses overlapping images and camera geometry to reconstruct three-dimensional ground measurements and terrain models.[4]
Words to Keep
similar triangles
Triangles with the same corresponding angles and proportional corresponding sides.
proportion
An equality between two ratios.
indirect measurement
Finding an unknown quantity by measuring other quantities connected to it by a known relationship.
photogrammetry
Measuring the position and shape of objects from images.
One Sentence to Keep
Similar triangles let us measure an unreachable height by comparing it with a smaller triangle we can measure directly.
What Should We Ask Next?
Shadows let us turn sunlight into triangles.
But what if the object is not on the ground?
The stars are far beyond any measuring tape. Ancient astronomers had to turn the sky itself into something measurable.
How Did Ancient Astronomers Turn the Sky into Angles?
Previous: How Did Eratosthenes Measure Earth with a Shadow?
Sources & Further Reading
- OpenStax, Contemporary Mathematics, “Triangles” — similar triangles and shadow-height applications.
- MacTutor, “Thales of Miletus” — later accounts of the pyramid-shadow story and historical cautions.
- NASA Johnson Space Center, “Photogrammetry” — definition and engineering applications of image-based measurement.
- U.S. Geological Survey, Structure from Motion Aided Photogrammetry — camera geometry, triangulation, and 3D reconstruction.
Historical note: the pyramid story about Thales survives only in much later sources. The worked example, algebraic notation, Python code, and modern photogrammetry connection are teaching tools, not reconstructions of one documented ancient calculation.