Temperature on the surface of the sun calculated with the Stefan-Boltzmann-rule In a German Wikipedia page, the following calculation for the temperature on the surface of the Sun is made:
$\sigma=5.67*10^{-8}\frac{W}{m^2K^4}$ (Stefan-Boltzmann constant)
$S = 1367\frac{W}{m^2}$ (solar constant)
$D = 1.496*10^{11} m$ (Earth-Sun average distance)
$R = 6.963*10^8 m$ (radius of the Sun)
$T = (\frac{P}{\sigma A})^\frac{1}{4} = (\frac{S4 \pi D^2}{\sigma 4\pi R^2})^\frac{1}{4}=(\frac{SD^2}{\sigma R^2})^\frac{1}{4} = 5775.8\ K$
(Wikipedia gives 5777K because the radius was rounded to $6.96*10^8m$)
This calculation is perfectly clear.
But in Gerthsen Kneser Vogel there is an exercise where Sherlock Holmes estimated the
 temperature of the sun only knowing the root of the fraction of D and R.
 Lets say, he estimated this fraction to 225, so the square root is about 
 15, how does he come to 6000 K ? The value $(\frac{S}{\sigma})^\frac{1}{4}$
 has about the value 400. It cannot be the approximate average temperature
 on earth, which is about 300K. What do I miss ?
 A: The relationship of temperature between a planet and a star based on a radiative energy balance is given by the following equation (from Wikipedia):

$T_p = temperature\ of\ the\ planet$ 
$T_s = temperature\ of\ the\ star$ 
$R_s = radius\ of\ the\ star$ 
$\alpha = albedo\ of\ the\ planet$ 
$\epsilon = average\ emissivity\ of\ the\  planet$ 
$D = distance\ between\ star\ and\ planet$
Therefore if Sherlock knows $\sqrt{\frac{R_s}{D}} = 0.06818$  and can estimate the Earth's temperature $T_p$ as well as $\alpha$ and $\epsilon$ then he can calculate the temperature on the surface of the sun which is the unknown variable $T_s$. 
Both $\alpha$ and $\epsilon$ have true values between zero and one. Say Sherlock assumed $\alpha = 0.5$ and $\epsilon = 1$ (perfect blackbody). Estimating the temperature of the Earth $T_p$ to be 270 K and plugging in all the numbers we have:

Which is very near the true average temperature of the surface of the sun, 5870 K. Case closed!
A: A rough estimate of a body's temperature in the solar system is 
$$T=\frac{280K}{\sqrt{D_{AU}}}$$
if we calculate the AU fraction from the Sun's "edge" to its center, R over D =  $4.65x10^-3$, and substitute this into the formula, the Sun's temperature would be about 4100K.
Not very close to your 5776 K, but utilizes the square root of the R D fraction.
The formula reflects effective temperatures. However peak, so called sub-solar  temperatures, are $\sqrt{2}$ times effective temperatures, which would yield about 5800K.
Clever Sherlock! 
