# View factor in concentric spheres

Consider a hollow sphere $$A$$ of radius $$1$$, containing a concentric sphere $$B$$ of radius $$r < 1$$. I am trying to calculate the view factor $$F_{A \rightarrow B}$$, but different approaches give me different results.

### Approach 1

The apparent radius of $$B$$, as seen from any point of $$A$$, is: $$\theta = \arcsin(r)$$.
The solid angle occupied by $$B$$ from that point of view is: $$\Omega = 2\pi(1 - \cos(\theta))$$.
The point of $$A$$ emits radiation in every directions uniformly across the hemisphere, so the fraction that hits $$B$$ is: $$\Omega / 2\pi = 1 - \cos(\arcsin(r))$$.
Since this is true of all points of $$A$$, we have $$F_{A \rightarrow B} = 1 - \cos(\arcsin(r))$$

### Approach 2

The areas of $$A$$ and $$B$$ are $$A_A = 4\pi$$ and $$A_B = 4\pi r^2$$, respectively.
Since B is convex, we have $$F_{B \rightarrow B} = 0$$ and therefore $$F_{B \rightarrow A} = 1$$.
By the reciprocity theorem, we have $$A_A F_{A \rightarrow B} = A_B F_{B \rightarrow A}$$ and therefore $$F_{A \rightarrow B} = r^2$$

Obviously these results are different. I assume the theorem is correct and my first approach is flawed. Where did I go wrong?