Every right triangle keeps a quiet promise.

Meet the three sides.

Pythagoras' theorem only works on one special kind of triangle, so let's make sure you can spot it. A right angle is a perfect square corner — exactly 90 degrees, the kind you see in the corner of a book, a window, or a piece of paper. A triangle that has one of these square corners is a right-angled triangle, and that corner is the star of the whole show. (A triangle can have at most one right angle — try drawing two and you'll see the sides can never close up.)

The two sides that actually make the right angle are the legs. Think of them as the two arms of an L. The third side — the long one stretched across from the corner, closing the triangle — is the hypotenuse (say it "hy-POT-uh-noose"). Two things are always true about it: it's the longest side, and it sits directly opposite the right angle. If you can find the right angle, the side facing away from it is your hypotenuse, every single time. Tap the buttons below to light each part up.

We always call the two legs a and b, and the hypotenuse c. It doesn't matter which leg is which — the rule treats them as equal partners.

What does "the square on a side" mean?

The theorem is full of the word "square," so let's pin down what it means. To square a number is simply to multiply it by itself. So 5² ("five squared") is 5 × 5 = 25, and 3² is 3 × 3 = 9. That little raised 2 is the only new piece of notation you need on this whole page.

But here's the lovely part, and the reason it's called "squaring" at all: 5² is also the area of a real square whose sides are each 5 units long. The word does double duty — it's a number trick and an actual shape, and the two meanings are the same thing. Multiply the side by itself and you've counted exactly how many little unit squares tile the inside.

So whenever Pythagoras says "the square on a side," picture literally drawing a square that sits on that side, using the side as one of its edges. Its area is the side length multiplied by itself. Slide the handle below and count the unit squares as the side grows — that count is the side, squared.

Squaring turns a length into an area. That's the whole reason Pythagoras' rule is about squares and not the plain sides.

Three squares that always balance.

This is the heart of the whole idea, so take your time with it. We've drawn a real square on each of the three sides — green on leg a, teal on leg b, and gold on the hypotenuse c — and written each square's area right inside it.

Drag the round handles on the corners — or nudge the sliders — to stretch the two legs to any sizes you like. Watch the two leg-squares grow and shrink, and keep one eye on the balance bars underneath: green plus teal always lines up exactly with gold. That's a² + b² = c² happening live, for triangle after triangle, with no exceptions.

Pour the small squares into the big one.

You've watched the bars balance — but why do they? Let's make it impossible to doubt, with the famous 3, 4, 5 triangle. The square on the short leg is 9 little tiles (3 rows of 3). The square on the longer leg is 16 tiles (4 rows of 4). And the square on the hypotenuse has room for 25 tiles (5 rows of 5).

Now do the only sum that matters: 9 + 16. That's 25. So if you scoop up all the tiles from the two smaller squares and pour them into the big one, they fill it perfectly — not one tile spare, not one space short. The area of the two legs' squares literally is the area of the hypotenuse's square; they're just the same amount of "space" arranged in different shapes. Press the button and pour them in.

This is the real reason the squares balanced in the demo above. The two leg-squares aren't just "close" to the big one, and it isn't a lucky accident of the numbers 3, 4 and 5 — the areas are the very same amount of space, every time. With other triangles the tiles wouldn't land in neat rows, but the total area always matches to the last sliver. Whole-number triples simply let you count it on your fingers.

If you can believe that two piles of tiles fill a third pile exactly, you already believe Pythagoras' theorem.

Finding the hypotenuse.

Now for the part that makes this rule genuinely useful. Imagine you can measure the two legs but the hypotenuse is awkward — it's a diagonal across a room, or the slope of a hill. You don't need to reach it at all. If you know both legs, the hypotenuse is no longer a mystery.

The recipe has three steps: square each leg, add the two answers together, and then take the square root. A square root just asks the reverse question — "what side length would give this area?" — so it perfectly undoes the squaring and hands you back a plain length. Set the two legs below and follow the working line by line; notice how the equation walks itself down to a single number.

Sometimes the answer is a tidy whole number, like the 6-8-10 you can try above. More often it's a decimal that runs on forever without repeating — take the simplest case of all, a right triangle with two legs of length 1: its hypotenuse is √2, which is about 1.41421… and never settles down. That's not the maths failing you. The diagonal of a square genuinely is that long; it's a perfectly real, exact distance that no decimal can ever finish writing out. Pythagoras is how humans first bumped into numbers like that.

So a "messy" hypotenuse is a feature, not a bug. The square root gives you the true length; rounding it just makes it convenient to carry around.

Finding a missing leg.

The rule is happy to run backwards, too. Suppose you know the hypotenuse and just one of the legs — the slanted distance and one straight one — and you want the missing leg. The same equation gets you there, but with one swap: this time you subtract instead of add.

Think it through with a² + b² = c². If you already know c and a, then b² must be whatever's left after you take a² away from c². So: square the hypotenuse, take away the square of the leg you know, and the square root of what remains is your answer. Drag the sliders and watch the subtraction do the work.

One rule of physics for this one: the hypotenuse must be the longest side. If you make the leg longer than the hypotenuse, there's no triangle to solve.

When all three sides are whole numbers.

You may have noticed that most right triangles have a hypotenuse with a messy decimal that never quite ends. That's normal — beautiful, even — and we'll come back to it. But a special, rare few have three whole-number sides that fit a² + b² = c² perfectly, with no decimals in sight. These tidy trios are called Pythagorean triples.

The most famous is 3-4-5: nine plus sixteen makes twenty-five, exactly. The next is 5-12-13. Builders and surveyors have leaned on these for thousands of years because whole numbers are so easy to measure out with a rope or a tape. Tap each triple below to check, with your own eyes, that the squares really do add up.

A fun fact to carry: 3-4-5 doubled is 6-8-10, and that works too (36 + 64 = 100). Any triple, scaled up, stays a triple.

Why anyone actually needs this.

It would be easy to think a two-thousand-year-old rule is a museum piece. It isn't — it's the quiet maths humming behind a surprising amount of everyday life. The trick is to spot the hidden right angle: anywhere a square corner meets a slanted distance, Pythagoras is already doing the work, whether anyone notices or not.

The ladder is the classic. It leans against a wall, making a right angle where the wall meets the ground. The ladder itself is the hypotenuse, and its length never changes — so as you slide its foot away from the wall, the height it can reach must shrink to keep the equation true. Slide the foot and watch the height fall exactly as Pythagoras predicts; it's also why a ladder placed too far out feels so unsafe.

Notice how often the same hidden triangle shows up. The walk home, the diagonal of a screen, the reach of a ladder — each is really just two known straight distances and one slanted one you'd rather not measure. Once you start looking, you'll catch right triangles hiding in staircases, phone screens, sports pitches, even the shortest path a video-game character takes across a grid. Pythagoras is the bridge between "how far across and how far up" and "how far in a straight line."

That builder's trick is the theorem run in reverse: if the three sides happen to fit a² + b² = c², then the corner must be a right angle. Sides and angles keep each other honest.

"Does it work for any triangle?"

This is the trap almost everyone falls into at least once, so let's clear it up now. The answer is a firm, friendly no. Pythagoras' theorem only holds when the corner is exactly a right angle — 90 degrees, not a degree more or less. The instant you stretch or squeeze that corner, the spell breaks and a² + b² is no longer equal to c².

The demo below makes the point honestly. The two legs stay locked at 4 and 3, so a² + b² stays parked at 25 the whole time. But you can swing the corner open and shut, and you'll see c² wander above and below 25 — meeting it at one single, magical setting. If your triangle isn't right-angled, don't reach for this rule; it simply won't be true.