 These three squares were made of gold, and I offered you either the large square or the two smaller squares, which would you choose? You may be surprised to hear that the two smaller squares are the exact same size as the larger square, so whichever you chose, you'd have received the same amount of gold. In this video, we're going to look at the relationships between the squares on the sides of right-angled triangles. Thousands of years ago, maybe even 4,000 years ago, it was discovered that for a right-angled triangle, the square on the biggest side is equal to the sum of the squares on the other sides. This is known as Pythagoras' theorem. It can be written like this or like this. It just depends which side you label as A, B, and C. But why is it useful? If you know the lengths of two sides of a right-angled triangle, you can really easily find the length of the other side. So let's have a look at an example. This is where Pythagoras' theorem comes in handy. We can label our sides as A, B, and C. And using A squared equals B squared plus two squared, substitute in the numbers, and we get four squared plus three squared. So A squared equals 16 plus nine, which is 25. Now be careful, the answer isn't 25. This is just A squared, the size of the whole square. We just want A. We need to square root the answer, and so A equals five. See how easy it is? Give this question a go yourself. Pause the video, work out the answer, and click play when you're ready to check. Did you get it right? You should have got 24.5 centimeters. If you did get it right, and you want to skip the explanation, click here. Otherwise, let's go through it together. Start by labeling the longest side as A, and then B and C are the short sides. And using Pythagoras' theorem, A squared equals five squared plus 24 squared, and we get A squared equals 601. Remembering to square root, so square root 601, and we get A equals 24.5 centimeters. One more question for you to do. It's just a little bit different because this time, we're looking for a short side, not a long side, so we just need to rearrange Pythagoras. Pause the video, work it out, and click play when you're ready to check. Did you get it right? You should have got three kilometers. If you got it right and want to skip the explanation, click here. Otherwise, let's go through the working together. Start by labeling the longest side as A, and then B and C are the short side. So in this example, we already know the longest length, so we'll just use a rearranged version of Pythagoras' theorem. The biggest square, take away a smaller square, will give you the other smaller square. Substituting in our numbers, 3.5 squared, take away 1.8 squared, equals C squared. So C squared is 9.01, and square-rooting that, we get C equals 3 kilometers. So all you need to know about Pythagoras' theorem is that it is only for right-angled triangles. The squares of the smaller sides add up to equal the square of the bigger side. And if you know the length of the bigger side, you square it and take away the square of the smaller side.