Another application would be laying out square angles. The theorem is able to make sure that buildings are square. A triangle whose side lengths correspond with the Pythagorean Theorem, such as a 3 foot by 4 foot by 5 foot triangle, will always be a right triangle. Construction workers will set out a triangle from three strings that correspond with square corners between the two walls within the area that they are working on. If the string lengths were measured correctly, the corner opposite the triangle's hypotenuse will be a right angle, so the builders will know they are constructing their walls or foundations on the right lines.
Another application would be navigation (Morris). The Pythagorean Theorem is useful for two-dimensional navigation. You can use it and two lengths to find the shortest distance. For example if you were in an airplane and you needed to navigate to a point that is 250 miles north and 500 miles west, you can use the theorem to find the distance from your plane to that point and calculate how many degress to the west of north you would need to follow to reach that point. The west and north distances would act as the legs. The same principles can be use for sea navigation. No more getting lost at sea!
One more application is surveying (Morris). Surveying is the process by which cartographers calculate the numerical distances and heights between different points before creating a map. Since terrain is often uneven, surveyors have to find ways to take measurements in a systematic way. The steepness of slopes of hills and mountains can be calculated using the Pythagorean Theorem. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope's line of sight and the measuring stick form a right angle. Since the surveyor knows both the height of the measuring stick and the horizontal distance of the stick from the telescope, he can then use the theorem to find the length of the slope that covers that distance, and from that length, determine how steep it is.
Below is one last example of the application of the Pythagorean Theorem. It is neat and eye opening, because a lot of the time we think the Pythagroean Theorem only involves triangles but it is much more than just triangles. Take a look at the site and my own original video below! If the video isn't working please visit the link directly below to vew it.