For a hemispherical furnace with a flat circular base of diameter D, the view factor from the dome to its base is

Concept:

Shape Factor is defined as the fraction of radiation energy leaving a surface that reaches another surface, for example, F12 means the fraction of energy leaving the surface 1 reaches surface 2.

Summation Rule: F11 + F12 = 1

Reciprocity theorem: A1 F12 = A2 F21 (A is the surface area)

Calculation:

Let us denote the hemishpherical body as body 1, and the flat surafce as body 2

From the summation rule F22 + F21 = 1, 

Since no radiation leaving the surface 2 will reach surface 2 again ∴ F22 = 0, F21 = 1

Now from reciprocity theorem,

A1 F12 = A2 F21

2 x π x r2 x F12 = π x r2 x1

∴ F12 = 0.5