Some approximations for $\arccos(1/(1+x))$

The Taylor series for the cosine is $$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dotsb.$$

Truncating this series after the $x^2$ term gives the rather good approximation $$\cos x \approx 1 - \frac12 x^2,$$ from which, by substituting $\sqrt{2y}$ for $x$, we get $$\cos \sqrt{2y} \approx 1 - y$$ and thus $$\arccos (1-y) \approx \sqrt{2y}.$$

Since, for small values of $y$, $$\dfrac{1}{1+y} = 1 - \dfrac{y}{1+y} \approx 1 - y,$$ it follows that $\sqrt{2y}$ is also a good approximation for $\arccos \dfrac{1}{1+y}$ when $y$ is small.

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To verify this approximation, consider the original equation $$d(h) = 2\pi R \cdot \arccos\frac{R}{R+h} = 2\pi R \cdot \arccos\frac{1}{1+\frac{h}{R}}. $$

Since $h/R$ is small in this case, $$d(h) \approx 2\pi R\sqrt{2\tfrac{h}{R}} = \pi\sqrt{8R} \cdot \sqrt{h} \approx 22441\sqrt{h}, $$ which only differs from Excel's calculation of the coefficient by $0.42\%$.