It's Just Measuring

In 240 BC, a librarian in Alexandria measured the circumference of the Earth. His equipment: a stick, a shadow, and the mathematics you are about to learn.

That sentence should bother you a little. A stick? To measure a planet? We will get there. But let me start with something you can picture more easily.

You are standing outside on a sunny day, next to a tall tree. You want to know how tall the tree is. You cannot climb it. You have no ladder, no laser rangefinder, no drone. What you do have: your own height, your own shadow, and the tree's shadow.

Here is the trick. The sun hits you and the tree at the same angle. Your height and your shadow form a triangle. The tree's height and the tree's shadow form a larger triangle — same shape, different size. If you are 6 feet tall and your shadow is 4 feet long, the ratio of height to shadow is 6 to 4, or 3 to 2. The tree's shadow is 20 feet long. Its height must be 30 feet. You measured the tree without touching it.

Think about what just happened. You measured three things you could reach — your height, your shadow, the tree's shadow — and mathematics told you a fourth thing you could not reach.

That is a different kind of measuring than holding a ruler up to something.

The interesting kind of measuring is exactly this: deducing what you cannot reach from what you can.

That is the entire subject. Geometry is the science of deducing measurements you cannot take directly from measurements you can. The tools are simpler than you might expect. By the end of this page, you will understand where those tools come from, and you will use them to do exactly what that librarian did 2,200 years ago.

You will measure the Earth with a stick.1

You just saw that a stick, a shadow, and a triangle can measure a tree. That raises a question: why a triangle? Of all the shapes available, why did the entire science of measurement get built around the three-sided one?

Why Triangles? #

Pick up three sticks. Lay them on the ground so their ends touch, forming a triangle. Now try to change the shape without breaking a stick.

You cannot. The triangle will not budge. Three sticks of fixed lengths lock into exactly one shape. There is no wiggle room, no freedom, no flex.

Now pick up four sticks of the same length and form a square. Push on one corner. What happens?

The square collapses. And it keeps collapsing — into a diamond, a thinner diamond, something that barely qualifies as a shape at all. Four sticks can form infinitely many quadrilaterals. The shape is not determined.

This is worth pausing on. Three sticks: one shape. Four sticks: infinitely many shapes. What changed?

The answer has to do with how many measurements it takes to pin down a shape completely. A triangle has three sides, and those three side lengths leave zero freedom — the shape is fully determined. A quadrilateral has four sides, but four side lengths leave one degree of freedom. It can still wobble. You would need an additional measurement — a diagonal, say — to lock it down. And that diagonal would split the quadrilateral into two triangles.2

Here is the deeper point. Take any shape at all — a pentagon, a hexagon, an irregular blob. You can always cut it into triangles. A quadrilateral splits into 2 triangles. A pentagon splits into 3. Before I state the general pattern, try a hexagon in your head. How many triangles?

Four. An n-sided polygon splits into exactly $n - 2$ triangles.

Try it yourself. In the interactive below, drag the vertices of a polygon into any shape you want. Watch it decompose into triangles. Then toggle the wobble mode: click a quadrilateral and watch it flex; click a triangle and watch it refuse.

Every polygon decomposes into triangles. Every curved boundary can be approximated by many small triangles. So if you can measure a triangle, you can measure anything. Triangles are the atoms of measurement.3

A triangle has six measurements: three sides and three angles. But you do not need all six. Because a triangle is rigid, three measurements (the right three) determine the other three. This is what makes triangles so useful: a small number of accessible measurements pins down everything else.

Which brings us to the most famous relationship between those measurements — and it starts, naturally, with a stick and its shadow.

The Pythagorean Theorem — Measuring by Deduction #

Go back to the stick and its shadow. The stick is vertical. The shadow lies flat on the ground. The line from the tip of the stick to the tip of the shadow cuts diagonally through the air. Three lines: the stick, the shadow, and the diagonal. They form a right triangle.

Suppose you know the stick is 3 feet tall and the shadow is 4 feet long. How long is that diagonal line?

Before I tell you, I want you to guess. That was an easy warm-up — let me give you a harder one. In the interactive below, a right triangle has legs of 5 and 12. Use the slider, and commit to a number for the third side. Then we will see why the answer has to be what it is.

The diagonal of the 3-4 triangle is 5. And the hypotenuse of the 5-12 triangle? It is 13. Not 17 (that would be the sum). Not 8.5 (that would be the average). Exactly 13. And the reason is not a formula — it is a picture.

Let me show you. Take four copies of your right triangle and arrange them inside a large square whose side length is $a + b$ (the sum of the two short sides). You can do this two different ways.

In the first arrangement, place all four triangles so their hypotenuses face inward, forming a tilted square in the center. The outer square has side $a + b$. The four triangles take up some of the area. The uncovered space in the middle is a single tilted square whose side length is the hypotenuse, $c$. Its area: $c^2$.

Now rearrange the same four triangles inside the same outer square. Pack them into two rectangular pairs — two triangles in the top-left and two in the bottom-right, flush against the edges. The uncovered space is now two smaller squares stacked in an L: one with side length $a$ (area $a^2$) and one with side length $b$ (area $b^2$). Total uncovered area: $a^2 + b^2$.

The outer square is the same size in both arrangements. The four triangles are identical. So the uncovered area must be the same. That means:

$$c^2 = a^2 + b^2$$

This is not an equation to memorize. It is a geometric fact you can see. One tilted square, or two axis-aligned squares. Same leftover area. That is why $5^2 + 12^2 = 13^2$. That is why, for any right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.4

Let that sink in. You knew two sides. The theorem gave you the third. You measured by deduction. No ruler touched the diagonal — mathematics told you it was there, and told you how long it was.56

The Pythagorean theorem relates one side of a triangle to the other sides. But the tree measurement from the introduction did not use side lengths at all. It used shadows and ratios. To understand why that worked, we need to understand what happens when two triangles have the same shape but different sizes.

Similar Triangles — The Measurement Multiplier #

Here is a question that sounds simple but turns out to be the key to everything: if you take a triangle and scale it up — make it bigger without changing its shape — what stays the same?

Take a right triangle where the short sides are 3 and 4. Now scale it up: sides of 6 and 8. Scale again: 30 and 40. Scale once more: 300 and 400. The side lengths are changing wildly. But before I say anything more, try the interactive below. Drag the triangle bigger. Drag it smaller. Watch the side lengths change. And look at the ratios between the sides.

Do you see it? The ratios are not moving. The triangle with sides 3 and 4 has a short-to-long ratio of 0.75. The triangle with sides 300 and 400? Also 0.75. The triangle got a hundred times bigger, and the ratio did not budge.

This is not a coincidence. Two triangles with the same angles must have the same ratios between their corresponding sides. We call such triangles "similar" — same shape, possibly different size. And the crucial fact is this: the ratios between their sides depend only on the angles, not on the scale.7

Why does this matter for measurement? Because it means you can build a small triangle that has the same angles as a big one, measure the small triangle's sides, compute the ratios, and those ratios apply to the big triangle too. You measure the model. The ratios carry the measurement to the real thing.

This is exactly what happened with the tree. Your height and your shadow formed a small right triangle. The tree and its shadow formed a large right triangle. The sun created the same angle for both, so the triangles were similar. The ratio of height to shadow was the same for you and the tree. You knew your ratio (6 feet tall, 4-foot shadow: ratio of 3 to 2). The tree's shadow was 20 feet. So the tree's height had to be 30 feet, because 30 to 20 is also 3 to 2.

Similar triangles are the tool that lets measurement scale. You cannot always reach the thing you want to measure. But if you can find a triangle with the same angles — often by exploiting a shared angle like the sun's elevation — then a ratio from a small, accessible triangle tells you the dimensions of a large, inaccessible one.

The ancient Greeks understood this deeply. Around 600 BC, Thales of Miletus reportedly measured the height of the Great Pyramid of Giza using exactly this method. He waited until the moment of day when his own shadow was exactly as long as he was tall — a 45-degree sun angle, making the height-to-shadow ratio exactly 1 to 1. At that moment, the pyramid's shadow length equaled the pyramid's height. He measured the shadow (from the center of the pyramid's base to the tip of the shadow, accounting for the pyramid's own width), and he had the pyramid.89

Try it yourself. In the interactive below, you can adjust the sun angle, your height, and the pyramid's shadow. Watch the similar triangles appear. Watch the ratios stay locked even as the scale changes.

But keep an eye on those locked ratios. Every time you change the angle, the ratios change accordingly. Every time you change the scale, the ratios stay put. Those ratios — the ones that depend only on the angle and never on the size — are doing all the measuring work in this essay. They will turn out to be important enough to deserve names. That is the next essay's job.

Measuring the Unmeasurable #

You are standing on the bank of a river. You want to know how wide it is. You cannot swim across with a tape measure. But you can do something cleverer.

Pick a landmark on the far bank — a distinctive rock, a tree stump. Call it point A. You are standing at point B, directly across from it. Now walk along your bank, perpendicular to the river. Keep walking until the landmark across the river is at exactly a 45-degree angle from the direction you walked. Stop. Call this point C.

Here is the beautiful part. You have just created a right triangle: the river's width is one leg (AB), your walk is the other leg (BC), and the angle at C is 45 degrees. A right triangle with a 45-degree angle has two equal legs — the height-to-base ratio is exactly 1 to 1. That is the same ratio Thales exploited with the pyramid. So the river's width equals the distance you walked.

If you walked 84 paces before the angle hit 45 degrees, the river is 84 paces wide. You measured a river without getting your feet wet.

Notice something different about this measurement. With the Pythagorean theorem, you started with two sides and deduced a third side. Here, you started with one side and one angle and deduced a second side. The angle carried measurement information. That shift — from sides-only to sides-and-angles — is what makes the next measurement possible. Because Eratosthenes did not measure a second side. He measured an angle.10

In the year 240 BC, the head librarian of the Great Library of Alexandria — a man named Eratosthenes — heard a curious report. In the city of Syene, far to the south, at noon on the longest day of the year, the sun shone directly to the bottom of a deep well. A vertical stick cast no shadow. The sun was exactly overhead.

Eratosthenes noticed something that most people would have ignored: on that same day, at that same time, a vertical stick in Alexandria did cast a shadow. A small one, but measurable. The sun's rays arrived at an angle of about 7.2 degrees from vertical — tilted just enough to cast a shadow you could see and measure.11

Let that settle in. Same sun, same moment, two cities. In one, the stick casts no shadow. In the other, the sun arrives 7.2 degrees off vertical. If the Earth were flat, the sun would hit both sticks at the same angle. The fact that the angles differ means the ground is curved. The Earth is a sphere.

But Eratosthenes did not stop at "the Earth is round." He measured how round.

Here is the key insight. The sun is enormously far away — so far that its rays arrive at the Earth essentially parallel, like rain falling straight down from a cloud the size of a continent. If the rays are parallel, then the 7.2-degree angle of the shadow in Alexandria is exactly the angle between the two cities as seen from the center of the Earth.

Now the proportion writes itself. If 7.2 degrees corresponds to the 500-mile distance between the two cities, then 360 degrees — the full circle — corresponds to the entire circumference of the Earth. Since $360 \div 7.2 = 50$, the circumference is $50 \times 500 = 25{,}000$ miles.

The modern value is 24,901 miles.

A human being, twenty-two centuries ago, with nothing but a stick, a shadow, and one proportion, measured the planet to within 2% accuracy.

Before I reveal the exact setup, I want you to try it. In the interactive below, set the shadow angle and the distance between the cities. Then guess the circumference. Drag the slider. Commit to a number. Then let the geometry show you why it has to be what it is.

$$\frac{7.2°}{360°} = \frac{500 \text{ miles}}{\text{circumference}}$$

$$\text{circumference} = \frac{360}{7.2} \times 500 = 25{,}000 \text{ miles}$$

The gap between the simplicity of the inputs and the enormity of the output is, I think, the most stunning thing in all of classical mathematics. One angle. One distance. The size of the Earth. The perspective we built across the first three sections — that triangles, their rigidity, and their proportional relationships are the atoms of measurement — is exactly the perspective that makes Eratosthenes' reasoning feel natural. An angle is a measurement. A proportion scales that measurement to the planet. That is all it takes.

And his instrument? A stick casting a shadow. The same setup we started with.

The Honest Boundaries #

You started this page knowing how to hold a ruler. You are ending it having measured the Earth.

The tools were simpler than they had any right to be. Triangles are rigid — three measurements pin down the shape. The Pythagorean theorem converts two accessible measurements into an inaccessible third. Similar triangles let ratios carry measurements across scale. That is the entire subject.

But I owe you some honesty about what we left out.

Everything we said assumes flat surfaces. On a curved surface — a sphere, a saddle shape — the rules change. Triangle angles no longer sum to 180 degrees. The Pythagorean theorem fails. The mathematics of curved surfaces is called differential geometry, and it is the language Einstein used to describe gravity. There is an irony here: Eratosthenes' measurement works because the Earth is curved, even though the geometry we used to derive it assumes flat triangles. The contradiction is only apparent. The triangles involved are small enough relative to the Earth that flatness is a good enough approximation. That "small enough to be flat" idea? It is exactly the "zooming in" principle from the calculus essay, applied at planetary scale.12

We also left something unnamed. In every section, we used ratios between the sides of triangles — and those ratios depended only on the angles. We used them to measure the tree. We used them to measure the pyramid. We used them to measure the river and the Earth. We relied on them completely, and we never once gave them names.

The next essay does exactly that. It turns out that naming those ratios — and building a systematic way to compute them for any angle — unlocks an entirely new set of measurement tools. Tools powerful enough to map the stars, navigate the oceans, and describe the oscillations at the heart of sound and light.

If geometry is about measuring what you cannot reach, the next essay is about the ratios that make the reaching possible.

That is the next essay: "It's Just Comparing."