Stretch a rope tight between two posts and it gives you a straight line. Stretch three ropes of length 3, 4 and 5 and you get something more useful: a perfect right angle, guaranteed, with no protractor in sight. Ancient builders knew this trick long before anyone wrote down why it works. The reason is one short equation — a² + b² = c² — and once it clicks, you can find a missing side of any right triangle, check whether a corner is truly square, and even measure the distance between two points on a map.
The cast: legs and a hypotenuse
Every right triangle has one corner that is exactly 90° — a square corner. The side directly across from that corner is special: it is the longest side, and it has its own name, the hypotenuse. The two shorter sides that meet at the right angle are called the legs.
Throughout this lesson we'll call the two legs \(a\) and \(b\), and the hypotenuse \(c\). The names \(a\) and \(b\) are interchangeable — it doesn't matter which leg you call which — but \(c\) is always the hypotenuse, the side facing the right angle.
The big idea is about areas, not lengths
Here is the part that most people are never shown, and it's the part that makes everything else obvious. The theorem isn't really about the lengths of the sides — it's about the squares built on those sides.
Imagine drawing an actual square outward from each side of the triangle. The square on leg \(a\) has area \(a^2\). The square on leg \(b\) has area \(b^2\). The square on the hypotenuse has area \(c^2\). The Pythagorean theorem says something beautifully physical:
In words — the area of the two smaller squares, added together, exactly equals the area of the big square on the hypotenuse. If you could pour the two small squares into the big one, they would fill it perfectly, with nothing left over and nothing missing.
Written symbolically, that's the equation you came for:
\[ a^2 + b^2 = c^2 \]Drag the legs of the triangle below and watch the three squares resize. Keep an eye on the numbers: the two small areas always add up to the big one.
With legs 3 and 4, the small squares have areas \(9\) and \(16\). Together that's \(25\) — and \(25\) is the square of \(5\). So the hypotenuse is exactly \(5\). No measuring required; the areas told us.
Finding the hypotenuse
When you know both legs and want the long side, rearrange the equation by taking a square root:
\[ c = \sqrt{a^2 + b^2} \]Square each leg, add the results, then take the square root. The order matters — you add before you root, because it's the areas that combine.
- Square each leg: \(3^2 = 9\) and \(4^2 = 16\).
- Add the two square-areas: \(9 + 16 = 25\).
- The hypotenuse is the side whose square is 25, so take the root: \(c = \sqrt{25} = 5\).
The hypotenuse is 5. This is the famous 3-4-5 triangle — the same one the rope-stretchers used.
Finding a leg
What if you already know the hypotenuse and one leg, and you want the other leg? The same equation works — you just rearrange it the other way. Start from \(a^2 + b^2 = c^2\) and subtract the known leg's square from both sides:
\[ b = \sqrt{c^2 - a^2} \]Notice it's now a subtraction inside the root. That makes sense: a leg is smaller than the hypotenuse, so you take away the small square from the big one to find what's left.
- Square the hypotenuse and the known leg: \(13^2 = 169\) and \(5^2 = 25\).
- Subtract — big square minus small square: \(169 - 25 = 144\).
- Take the root: \(b = \sqrt{144} = 12\).
The other leg is 12. (Check: \(5^2 + 12^2 = 25 + 144 = 169 = 13^2\) ✓)
Tip — add for the hypotenuse, subtract for a leg. If the side you want is the longest one, you're combining two squares, so you add. If the side you want is a leg, you're finding the leftover, so you subtract the known leg's square from the hypotenuse's square. Get the operation right and the rest is just arithmetic.
The converse: a test for right angles
So far we've used the theorem forwards: given a right triangle, the squares add up. The remarkable thing is that it also runs backwards. If you measure all three sides of any triangle and find that the two smaller squares add to the largest square, then the triangle must have a right angle. This is called the converse of the Pythagorean theorem.
That's exactly why the rope trick works. The builders didn't know the corner was square — they made it square by forcing the sides to be 3, 4 and 5, because \(3^2 + 4^2 = 5^2\). The lengths created the right angle.
In words — to test a triangle, square the longest side, then square the other two and add them. If the two sums match, it's a right triangle. If the two legs' squares come out bigger, the corner is sharper than 90°; if smaller, it's blunter.
Triples worth memorising
A Pythagorean triple is a set of three whole numbers that fit the equation perfectly. Because they avoid messy square roots, they show up constantly in textbooks and tests — so it pays to recognise a few on sight:
- 3, 4, 5 — since \(9 + 16 = 25\). The classic.
- 5, 12, 13 — since \(25 + 144 = 169\).
- 8, 15, 17 — since \(64 + 225 = 289\).
Any multiple of a triple is also a triple. Double the 3-4-5 and you get 6-8-10; triple it and you get 9-12-15. They're all the same triangle, just scaled — which is exactly what the next practice problem is built from.
One more gift: distance between two points
The theorem quietly powers something you'll use again and again in coordinate geometry. To find the straight-line distance between two points on a grid, treat the horizontal gap and the vertical gap as the two legs of a right triangle — the distance you want is the hypotenuse.
If you go across by 3 and up by 4 to get from one point to another, the direct distance is \(\sqrt{3^2 + 4^2} = \sqrt{25} = 5\). Same triangle, same theorem — now it's measuring a path across a map instead of a side of a shape. You'll meet this idea formally in a later lesson, but it's worth seeing that it's nothing new: it's Pythagoras, dressed for coordinates.
Practice
Try each one yourself, then reveal the full solution.
1. A right triangle has legs of length 6 and 8. Find the hypotenuse.
We want the longest side, so we add the two leg-squares.
Square the legs: \(6^2 = 36\) and \(8^2 = 64\).
Add them: \(36 + 64 = 100\).
Take the root: \(c = \sqrt{100} = 10\).
The hypotenuse is 10. (This is just the 3-4-5 triangle doubled.)
2. The hypotenuse of a right triangle is 13 and one leg is 5. Find the other leg.
We want a leg, so we subtract the known leg's square from the hypotenuse's square.
Square the hypotenuse and the known leg: \(13^2 = 169\) and \(5^2 = 25\).
Subtract: \(169 - 25 = 144\).
Take the root: \(b = \sqrt{144} = 12\).
The other leg is 12 — this is the 5-12-13 triple.
3. A triangle has sides 5, 12 and 13. Is it right-angled? Use the converse.
The converse says: if the two smaller squares add to the largest square, the triangle is right-angled.
The longest side is 13, so its square should be the total: \(13^2 = 169\).
Square the other two sides and add: \(5^2 + 12^2 = 25 + 144 = 169\).
The two sums match exactly: \(169 = 169\).
Yes — the triangle is right-angled, with the right angle sitting opposite the side of length 13.