= Solution
For the <sphere>, choose the inward normal so that the <second fundamental form> is $I/a$. Both principal curvatures are $1/a$, giving \b[$K=1/a^2$]; the outward choice reverses both signs and gives the same product.
The octant has area $4\pi a^2/8=\pi a^2/2$, so its curvature integral is $\pi/2$. Its sides are great-circle geodesics, hence have zero <geodesic curvature>. Each interior angle is $\pi/2$, so each exterior angle is also $\pi/2$. Therefore
$$
\boxed{\int_DK\,dA+\int_{\partial D}\kappa_g\,ds+\sum_i\theta_i=\frac\pi2+0+\frac{3\pi}2=2\pi,}
$$
verifying the polygon formula directly.
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