Please help finding the antenna temperature example from the book.

This is only an example from Kraus Antenna 3rd edition page 404.

Example 12-1.1 Mars temperature

The incremental antenna temperature for the planet Mars measured with the U.S. Naval Research Lab 15-m radio telescope antenna at 31.5mm wavelength was found to be 0.24K(Mayer-1). Mars subtended an angle of 0.005 deg at the time of the measurement. The antenna HPBW=0.116 deg. Find the average temperature of the Mars at 31.5mm wavelength.

The answer from the book:

\[T_s=\frac {\Omega_A}{\Omega_S}\Delta{T_A}\;\;\approx\;\;\frac{0.116^2}{\frac{\pi}{4}0.005^2}(0.24)=164K\]



I don't understand where the \[\frac {\pi}{4}\] in the denominator comes from. This is just a simple problem where the ratio of the area of two disk one with Θ=0.116/2 and the other with Θ=0.005/2. I try to use the following to calculate:

\[\Omega=\int_0^{2\pi} d\phi\;\int_0^{\theta/2}\sin {\theta} d \theta\]

So

\[\frac{\Omega_A}{\Omega_S}\;=\;\frac{\int_0^{2\pi} d\phi\;\int_0^{0.116/2}\sin {\theta} d \theta}{\int_0^{2\pi} d\phi\;\int_0^{0.005/2}\sin {\theta} d \theta}\;=\;\frac{2\pi\int_0^{0.116/2}\sin {\theta} d \theta}{2\pi\int_0^{0.005/2}\sin {\theta} d \theta}\]

\[ \frac{\Omega_A}{\Omega_S}\;=\;\frac{(1- 0.999999487)}{(1-0.999999999)}=538.2\]

But using the answer from the book:

\[\frac{0.116^2}{\frac{\pi}{4}0.005^2}\;=\;685.3\]

I just don't get the answer from the book. Can it be the limitation of my calculator to get cos 0.0025 deg? Can anyone use their calculator to verify the number? please help.

Thanks

Alan