quantum information - Why is $theta over 2$ used for a Bloch sphere instead of $theta$?

I'm a beginner in studying quantum info, and I'm a little confused about the representation of a qubit with a Bloch Sphere. Wikipedia says that we can use $$\lvert\Psi\rangle=\cos\frac{\theta}{2} \lvert 0\rangle + e^{i\phi}\sin\frac{\theta}{2} \lvert 1\rangle$$ to represent a pure state, and map it to the polar coordinates of the sphere. What I'm not sure about is, where does the "$\frac{\theta}{2}$" come in?

I mean, in polar coordinate, the vector equals $\cos{\theta}\ \hat{z} + e^{i\phi}\sin{\theta}\ \hat{x}$, but even if we use $\hat{z}=\lvert 0\rangle$ and $\hat{x}=\lvert 0\rangle + \lvert 1\rangle$, it's still different from above. How could this be transformed into the formula above?

Or... does this mean that the sphere is simply a graphical representation of $\theta$ and $\phi$, while $\lvert 0\rangle$ and $\lvert 1\rangle$ do not geometrically correspond to any vector on the sphere? (but here it writes $\hat{z}=\lvert 0\rangle$ and $-\hat{z}=\lvert 1\rangle$...)