What is Pythagoras's theorem?

Pythagoras's theorem is used to calculate the lengths of the sides of right-angled triangles. The theorem can be used to find the length of the third side of a right-angled triangle, as long as the lengths of the other two sides have been provided. The theorem is: a² + b² = c², where c is the length of the hypotenuse, or longest side. The hypotenuse always lies opposite to the right-angle, while a and b are the two sides which form the right-angle. It doesn't matter which of these two sides is a and which is b, as long as you are consistent in sums. Take, for example, the simplest question: a right-angled triangle has two sides of length 3 and 4 - these sides are adjacent to the right-angle. What is the length of the hypotenuse? Sides a and b are therefore 3 and 4. a² + b² = c²   so   3²+4² = c²     so   25 = c²   so  c = 5. The formula can also be rearranged. For example, if you are given the hypotenuse and one other side's length, you can still calculate the third side's length: A right-angled triangle has a hypotenuse of length 13 and another side of length 12. What is the length of the third side?  We know that c = 13 and another side - let's call it b - is 12. First, rearrange the formula to put it in terms of a: a² + b² = c²    becomes    a²  = c² - b² So a² = 13² - 12²    so  a² = 169-144    so  a² = 25  so  a = 5.