Surprising Uses of the Pythagorean Theorem
betterexplained.com
betterexplained.com
It really helps you visualize why you need to separate your lists into smaller piles before sorting if you can, since it takes the same amount of time to sort a 50 element list as it does sorting a 40 and a 30 both.
50^2 = 25^2 + 43.3^2
Strictly speaking, if you have a circle of radius 50, you can take the sine and cosine of any angle and get a possible way to split the length.
1) The sides of a triangle (3-4-5) can represent portions of any shape. For example, circle area (radius 3) + circle area (radius 4) = circle area (radius 5).
Instead of radius, you could pick diameter, circumference, or any line segment and the relationship would hold.
2) You can use the Pythagorean Theorem to split any squared quantity into two smaller ones. This can yield surprising insights; is it immediately clear that a list of 50 takes as long to bubble sort as a list of 30 and 40? =)