Why Does Doubling Occur in My Illuminance Calculation?

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Hello everybody,
this question is about an exercise but I post it here because it is not homework, it is an exercise that I've done to learn by myself. I hope it is ok.
It is an exercise from Irodov book (exercise 5.12)
The exercise says:
A small spherical lamp, uniformly luminous with radius R= 6.0 cm is suspended at an height h of 3 metres above the floor;
The luminance of the lamp is L=2.0*10^4 cd/m2, indipendent of direction.
Find the illuminance of the floor directly below the lamp.

the solution that the book (and my teacher) gives is

$$I= \pi\frac{R^{2}}{h^{2}} L = 25.13\, \, lux$$


I tried to solve it in this way:
the symmetry of the problem gives us many advantages; we can obtain the total luminous flux of the lamp by multiplying the luminance by the total surface and by 2 pi steradians (half of the maximum solid angle, because i assume the sphere doesn't radiate inside itself)

$$F=L (2 \pi) (4 \pi R^{2} )= L (8 \pi^{^2} R^{2} )$$

then, to have the illuminance of the floor just below the lamp, we can divide the total flux by the area of the sphere having radius h:

$$I= L (8 \pi^{^2} R^{2} )/ (4 \pi h^{2})= 2 \pi L \frac{R^{2}}{h^2}=50.27\; lux$$

but as you can see it is exactly twice the solution given by the book.
I suppose i am wrong, but i cannot understand why. can you help please?
Thank you in advance, sorry for my english

 

[member 553083]

.Your formula is correct, but the reason why you get twice the results of the book is because the luminance you are given is already the one of the hemisphere facing downwards. The total luminance of the sphere should be twice that value.