578. Check the Pythagorean theorem for a right triangle with legs 1 and 2.
$\triangleright$ The squares constructed on the legs have areas of 1 and 4. It remains to find the area of the square constructed on the hypotenuse. This can be done in two ways. First, it can be cut into 4 triangles and a central square. The triangles are equal to the original (legs 1 and 2) and have an area of 1. The area of four such triangles is 4, plus the unit square in the center, totaling 5, as required.

Another way to calculate the area of the square constructed on the hypotenuse is to surround it with triangles on all four sides, completing it to a larger square. By adding four triangles equal to the original, we get a square $3 \times 3=9$, thus the desired area is $9-4=5 . \triangleleft$