If you had to pick the single most famous formula in all of middle school math, it's probably a²+b²=c². When I first learned this formula, I just memorized it because it was going to be on the test — I only really understood it once I later saw, in a picture, what the equation actually means.
Here's the core idea. In a right triangle, call the two legs that form the right angle a and b, and call the longest side (the hypotenuse) c. Draw a square on each of the three sides, and the areas of the two smaller squares, added together, come out exactly equal to the area of the larger square. Take the 3-4-5 triangle as an example: 3×3=9 and 4×4=16, and adding them gives 25 — which lines up perfectly with 5×5=25.
What makes this theorem special is that there are dozens of different ways to prove it. Some cut up a picture and rearrange the pieces to show the areas are equal; others arrange four triangles inside a square to show the same thing. On our site, just moving the same four triangles around shows, through animation, how the area that looked like c² splits back into a² and b² — and seeing it happen with your own eyes builds the conviction "oh, it really is the same area" much faster than any explanation.
The Pythagorean theorem is useful outside the classroom too. It's the reason a carpenter can use a 3-4-5 tape-measure ratio to check whether a wall is standing at a true right angle, and this same formula is hiding behind calculating the straight-line distance between two points on a map, and even behind how a TV or monitor's screen size gets measured.
What's interesting is that the "converse" of this theorem holds too. In other words, if a triangle's three side lengths satisfy a²+b²=c², that triangle is guaranteed to be a right triangle — no protractor required. This principle has actually been used since ancient times: in ancient Egypt, people are said to have tied knots in a rope at regular intervals to make a 3:4:5 triangle, using it to precisely lay out the right-angle corners of pyramids and farm fields. Producing a perfect right angle with nothing but a simple rope, at a time when surveying technology was nowhere near as advanced as today, was a genuinely clever idea. Knowing that carpenters and gardeners still use a similar principle today to check the angles of walls or fences makes it feel oddly remarkable that math from thousands of years ago is still alive and in use.
Here's a useful tip: combinations like 3-4-5, 5-12-13, and 8-15-17, where all three sides come out to whole numbers, are called "Pythagorean triples," and memorizing a few of them makes calculations much faster when solving problems. Kids tend to quietly enjoy it when you turn finding these triples into a quiz. Start with problems that give two sides and ask for the third, and once that feels comfortable, move on to the reverse — giving three side lengths and asking "is this a right triangle?" — which deepens the thinking a level further. Practicing back and forth between the two directions naturally teaches you that the same formula works both as "a tool for finding a missing side" and as "a tool for checking a triangle's properties."