Sometimes when turning a corner you have walk through 90 degrees or crossing a rectangular field you'd take the shortest distance across it which is its hypotenuse.
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While doing your homework, or on mapping, or for distance.
If you are packing items into a box, you can use square roots along with Pythagorean theorem to help determine if a smaller box could work with items packed diagonally.
In comparing the effect of focal lengths in digital cameras with different sized sensors.
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