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Lesson 3 · Geometry

Pythagoras' Magic

A 2,500-year-old rule that's never wrong. Find a right-angle triangle anywhere in the world — this rule still works.
Scene 1

A right-angle triangle has three sides 📐

A right-angle triangle is one with a square corner — like the corner of a book. The two short sides we'll call a (pink) and b (teal). The long side is called c (gold) — it's always the longest.

Now here's the magic. Build a square on each side, like little rooms. The pink square has area a × a = a². The teal square has area b × b = b². The gold square has area c × c = c².

Scene 2 · Try it

Drag the sliders. Watch what happens. ✨

Change the lengths of a and b. The pink and teal squares will grow or shrink. So will the gold one. Look at the numbers. The pink area plus the teal area always equals the gold area.

a = 3
b = 4
9 + 16 = 25
+ =   ·   so c = 5
Scene 3

Why does anyone care? 🤔

This rule lets you measure things you can't reach. How tall is a tree? Walk away from it, measure your shadow, and use Pythagoras. How far is the airport from your house in a straight line, even if the road bends? Pythagoras.

Pilots, builders, video-game makers, and even your phone's GPS use this exact rule millions of times a day. It was discovered in ancient Greece — but the universe was already obeying it long before that.

The Big Idea

Some rules are forever 🌟

+ =

This is one of the oldest and most reliable facts humans know. It works on Earth, on Mars, in another galaxy, billions of years ago, billions of years from now.

Pythagoras lived 2,500 years ago. He's long gone, but his rule will still be true 2,500 years from now. That's the power of math.