# Mistakes about illuminance

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 simmetry 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