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 does not mean the shadow somehow contains the height.
It means the Sun has built a smaller triangle and a larger triangle with the same shape.
Quick Answer
Put a short vertical object near the tall object. Measure the short object's height and shadow, then measure the tall object's shadow under the same sunlight direction.
The two height-and-shadow triangles have the same two angles: a right angle and the same sunlight angle.
That makes them similar triangles, so corresponding sides share the same scale factor.
tall height / tall shadow = short height / short shadow
Indirect measurement works when an unknown length belongs to the same geometric pattern as lengths we can measure directly.
The Structural Question — How Can a Known Shape Transfer a Measurement?
The previous article used a local angle and a baseline to infer Earth's circumference.
This time the hidden quantity is smaller: the height of a tower, tree, or wall.
But the architecture is the same:
reachable measurements + invariant relationship → unreachable quantity
The invariant relationship here is shape. One triangle is larger than the other, but the scale between corresponding sides is consistent.
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. Two equal corresponding angles are enough to establish similarity; this is the AA similarity idea. Similar triangles have proportional corresponding sides.[1]
Figure 1. Same sunlight angle gives the two right triangles the same shape.
The Whole Method Is One Scale Factor
h = height of the reference 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
Or think in terms of scale:
scale factor = S / s
H = h × scale factor
If the tower's shadow is 20 times the stick's shadow, the tower is 20 times the stick's height.
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.
Why Must the Measurements Use the Same Sun Angle?
Because the sunlight angle is what makes the two triangles the same shape.
As the Sun moves across the sky, the object's height stays fixed but its shadow length changes.
If the reference shadow is measured at one time and the tower shadow much later, the triangles may no longer share the same acute angle, so the simple proportion may fail.
For nearby objects, measuring both shadows at essentially the same time is the simplest way to keep the model consistent.
What If the Ground Is Sloped?
The simple ratio assumes both shadows lie on compatible level ground and both objects are vertical.
On sloped or uneven ground, the raw distance measured along the ground is not automatically the horizontal side of the ideal right triangle.
The situation can still be corrected with more geometry, but the one-ratio formula is no longer automatic.
A proportion is not magic. It works because the physical setup matches the geometric model.
Why Does a Very Short Shadow Make Measurement Harder?
Suppose the reference shadow is only 10 cm long. An endpoint error of 1 cm is already 10% of that measurement.
If the shadow is 1.0 m long, the same 1 cm error is only 1%.
This is a simple example of relative error: the same absolute mistake matters more when the measured quantity is small.
Can We Do It Without a Reference Stick?
Yes—if we can measure the Sun's elevation angle directly.
For a vertical object on level ground:
tan(α) = H / S
so:
H = S tan(α)
The reference-stick method avoids measuring α directly. The smaller triangle already carries the same angle, so h/s plays the role of tan(α).
This is a natural bridge from similar triangles to trigonometry.
A Story About Thales — Useful, but Historically Uncertain
A famous tradition credits Thales of Miletus with measuring an Egyptian pyramid from its shadow.
But the surviving accounts were written centuries after Thales, and they do not all describe the same method.
One tradition says he observed the pyramid when a person's shadow equaled that person's height. Another later account describes a stick and two proportional triangles, much closer to the general method used here.[2]
So it is reasonable to teach the geometry—but not to pretend we know the exact historical procedure with certainty.
The geometry is solid. The exact story of how Thales used it is less certain.
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 and compatible for both shadows.
- 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.
Build a Better Field Experiment
- Use a reference object whose height is known accurately.
- Check that the reference object and target are vertical.
- Choose reasonably level ground.
- Measure both shadows at essentially the same time.
- Mark the shadow tips before they move.
- Repeat the measurements and compare the estimates.
If repeated estimates disagree, that disagreement is information. It tells you the measurement process needs attention.
Where This Idea Lives Today
Modern surveyors use much more precise instruments than shadow sticks.
Cameras and computers can also recover sizes and positions of objects that are hard to touch directly.
NASA defines photogrammetry as determining the position and shape of objects from two or more images acquired from different perspectives.[3]
Photogrammetry is not the same calculation as the shadow method. It may use camera calibration, perspective geometry, matching image features, and triangulation.
But the deeper strategy is closely related:
Measure a projection you can observe, use a geometric model, and recover a size or position you cannot directly touch.
NASA uses photogrammetry for engineering measurements including three-dimensional position, trajectory, structural motion, surface damage, and alignment.[3]
Modern USGS Structure-from-Motion workflows similarly use overlapping imagery and camera geometry to reconstruct three-dimensional point clouds and terrain products.[4]
Why This Story Matters
The shadow trick is memorable because the equipment is simple. But the transferable idea is more important than the trick.
We do not always need direct access to the quantity we want.
If we can measure something connected to it by a reliable relationship, geometry can carry the information across.
direct access is optional; a trustworthy relationship is essential
Words to Keep
similar triangles
Triangles with the same corresponding angles and proportional corresponding sides.
AA similarity
A triangle-similarity condition: if two corresponding angles are equal, the triangles have the same shape.
scale factor
The multiplier that converts corresponding lengths in one similar figure to another.
proportion
An equality between two ratios.
indirect measurement
Finding an unknown quantity by measuring other quantities connected to it by a known relationship.
relative error
The size of a measurement error compared with the size of the quantity being measured.
photogrammetry
Recovering geometric information about objects or scenes from photographs or other images.
One Sentence to Keep
Similar triangles let a measurable small triangle transfer its scale to a larger triangle whose height we cannot reach 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.
Next: 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.
- Mathematical Association of America, “Some Original Sources for Modern Tales of Thales — The Tale of the Pyramids” — comparison of later ancient accounts of the pyramid-shadow story and their historical limitations.
- MacTutor, “Thales of Miletus” — historical caution about later attributions to Thales and the multiple pyramid-shadow traditions.
- 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.
Research note: Reddit, YouTube, and Mathematics Stack Exchange were used as question sensors for recurring reader confusion—especially why the triangles are really similar, whether the measurements must be simultaneous, what happens on sloped ground, whether a reference stick is necessary, and how measurement error enters. They were not used as factual authorities.
Historical note: the pyramid story about Thales survives through much later sources, and different versions describe different methods. The worked example, tangent formula, Python code, field-test advice, and photogrammetry connection are modern teaching tools.