Area of a small geodesic polar ball under an upper curvature bound
= Area of a small geodesic polar ball under an upper curvature bound
If $K\leq C$ with $C>0$, scalar comparison gives
$$
h(r,\theta)\geq\frac{\sin(\sqrt C r)}{\sqrt C}
$$
before $\pi/\sqrt C$. Consequently
$$
\operatorname{Area}B(p,\varepsilon)
\geq\frac{2\pi}{C}\bigl(1-\cos(\sqrt C\varepsilon)\bigr)
=\pi\varepsilon^2(1+O(\varepsilon^2)).
$$