In spherical polar co-ordinate system why the polar angle ranges from $0$ to $\pi$? In spherical polar co-ordinate system why the polar angle ranges from $0$ to $\pi$? Why not $2 \pi$? Why the use of $2\pi$ would count every point twice? Please explain. Thanks in advance. 
 A: Look at the coordinate system shown below.

To span the whole of space if $\theta$ goes from $0$ to $2\pi$ then $\phi$ only has to go from $0$ to $\pi$.
If the range of $\phi$ is $0$ to $2\pi$ then each point is counted twice.  
The ranges of $\theta$ and $ \phi$ could be interchanged but by convention this is not done.
A: I see some confusion... normally the azimuthal angle can range from 0 to $2\pi$, while the polar angle ranges from 0 to $\pi$.
Anyway, let's consider $\theta$ as polar angle (or colatitude), and $\phi$ as azimuthal angle (or longitude, if you consider a geographic reference frame).
Imagine you're on Earth: a meridian starts at North Pole (0°) and it ends at South Pole (180°). To obtain the whole Earth (i.e. the sphere) from just one meridian, you need to rotate it around the polar axis of 360°, or $2\pi$. If you rotate less than that $2\pi$ you'll not obtain the whole sphere, otherwise you'll have a superimposition. I find it easier to understand if you think of the geographical reference frame (it's kind a visual thing).
If you think in mathematics, you will have two angles, already defined above as $\theta$ and $\phi$, with the same meaning. For avoiding superimpositions you'll have $0\le \theta \le \pi$ (with $\theta$ polar angle) and $0\le \phi \le 2\pi$ (with $\phi$ azimuthal angle). It's a convention that states that your azimuthal angle ranges in this interval: you could invert the notation (polar angle ranging from 0 to $2\pi$) but it's not wise.
If you want to know more, here it's quite complete: http://mathworld.wolfram.com/SphericalCoordinates.html
