In Wikipedia https://en.wikipedia.org/wiki/Curvature
There are two formulas for the curvature of a planar curve:
1) parametrically
For a plane curve given parametrically in [[Cartesian coordinates]] as $ c(t) = ( x(t),y(t)) $ the curvature is
$$ \kappa = \frac{|x'y''-y'x''|}{\left(x'^2+y'^2\right) ^ \frac32}$$
where primes refer to derivatives $ \frac{d}{dt}$ with respect to the parameter $t$.
And 2) Curvature from arc and chord length
Given two points $P$ and $Q$ on $c$, let $ s(P,Q) $ be the arc length of the portion of the curve between $P$ and $Q$ and let $ d(P,Q) $ denote the length of the line segment from $P$ and $Q$. The curvature of $c$ at $P$ is given by the limit
$$ \kappa(P) = \lim_{ Q \to P} \sqrt \frac{24 \big(s(P,Q) -d (P,Q)\big)} {s(P,Q)^3}$$
Where the limit is taken as the point $Q$. approaches $P$ on $c$. The denominator can equally well be taken to be $ d(P,Q)^3$. The formula follows by verifying it for the osculating circle.
(my remark: really ?? I did not manage this)
How can the second formula be derrived from the first one? (or even from the osculating circle?)
You may even assume the curve $c$ is a circle with radius $r$
The formula could be verified for the osculating circle as follows. Considering two points $P$ and $Q$ at the extremities of given circular arc of the osculating circle, subtended by an angle $\alpha$, the formula reads:
$$ \kappa (P) = \lim_{Q \to P} \sqrt{ \frac{24 \Big( s(P,Q) - d(P,Q)\Big)}{s(P,Q)^3}} = \\ \lim_{\alpha \to 0} \sqrt{ \frac{24 \Big( \alpha r - 2r \sin{\frac{\alpha}{2} }\Big)}{\alpha^3 r^3}} = \\ \lim_{\alpha \to 0} \sqrt{ \frac{24 \Big( \alpha r - 2r (\frac{\alpha}{2} -\frac{1}{3!}({\frac{\alpha}{2})^3+\dots} \Big)}{\alpha^3 r^3}} = \\ \sqrt{ \frac{24 \Big( r \frac{\alpha^3}{24} \Big)}{\alpha^3 r^3}} = \frac{1}{r} $$
where the Taylor expansion $\sin x \approx x - \frac{x^3}{3!} + \dots$ is exploited.
As the linked Wikipedia article points out, the denominator could be substituted by $d(P,Q)^3$ and the same results follows, as it is easy to verify from the above calculation.