Antie, a smart ant living on a smooth surface $S$ of $\mathbb{R}^3$, would like to evaluate the Gauss curvature $K$ at a certain point $P\in S$. Antie is aware of Gauss Theorema Egregium, according to which Gauss curvature may be evaluated using the first fundamental form of the surface. So, Antie grabs a ruler and a protractor, determines a neighbourhood around $P$ and is ready to start calculating, since it has heard that the first fundamental form is related to lengths and angles around $P$ (words like tangent space do not really make sense to Antie). Αntie knows that if he knew some quantities, called $E$, $F$, $G$, then it would be able to evaluate $K$.
But of course Antie is not aware of the way the surface is embedded in space $\mathbb{R}^3$ and its local parametrization $\sigma(u,v)=( \sigma_1(u,v), \sigma_2(u,v),\sigma_3(u,v))$ around $P$ (in order to evaluate $E=\sigma_u\cdot \sigma_u$ etc around $P$).
How would we help Antie calculate all the quantities needed for Gauss curvature $K$ at $P$?
Assumption: the “ruler and protractor” has the following functions:
Ruler: draw a geodesic between two points, and measure the length of that geodesic
Protractor: given two geodesics intersecting at a point P, measure the angle they make at point P.
At least on the plane, these are the correct properties of the ruler and the protractor. I don’t know if they are the correct properties of these instruments on the surface.
I don't know a lot of differential geometry, but here might be a feasible approach.
Draw a sufficiently small geodesic right triangle $ABC$ around $P$, say with angle $B$ being 90 degrees and $AB = BC$, and (very accurately) measure the angles $\angle ABC, \angle ACB, \angle BAC$. By Gauss-Bonet, we have $$\int_{\triangle ABC} K dA = \angle ABC + \angle ACB + \angle BAC - \pi.$$ Now, if $K_P$ is the curvature at point $P$, we have $$\int_{\triangle ABC} K dA = (1 + o(1)) \text{Area}(\triangle ABC) \cdot K_P$$ and $$\text{Area}(\triangle ABC) = \frac{1 + o(1)}{2} AB \cdot BC.$$ So, as the triangle's area converges to $0$, we have $$K_P \sim \frac{2(\angle ABC + \angle ACB + \angle BAC - \pi)}{AB \cdot BC}.$$ The ant can measure all the quantities on the right hand side, so by taking sufficiently small triangles, can measure $K_P$ with arbitrary accuracy.