Calculate the shadow on earth of a large orbital disk How to calculate the shadow on earth of a large orbital disk (low orbit)?
It might be easier to use similar triangles. 

$$\frac{Rd}{b} = \frac{Rs}{a}$$

$Rs$ is 696,000 km and $Rd$ is 5 km. $a$ is about 150,000,000 km and c is 80 km.

The the smallest triangle is used next:

$$\frac{r}{b-c} = \frac{Rs}{a}$$

Solve for r and you get

$$r = Rd - Rs \frac{c}{a} = \text{4.63 km}.$$

Extra digits are not helpful because the exact diameter of the Sun depends on how you define it and the distance from the Sun to the Earth varies by almost +/-2 percent.

$Rs/a$ is about 0.00464 and that's also the half-angle of the Sun in radians. Convert it to degrees my multiplying by 180/pi and you get 0.266 degrees, or a quarter of a degree. The full diameter of the Sun is double that, or about a half degree.