Clever Approximations

Computing arc length and generalizing the trick

Suppose you have a circular. You measure the chord of the arc, and the chord of half the arc. It may be surprising that you can find the length of the arc from these two measurements. It’s even more surprising that there is a very accurate approximation for the arc length that doesn’t involve any trig functions. Details here.

The trick that makes the arc length approximation work cancels out the second order terms in series that are already missing first and third order terms. I apply the same idea to a different problem here.

Have fun.