In this post: calculating volume of sphere with integration, the OP states that the equation of a sphere is $\sqrt{a^2 - x^2 - y^2}$. I already know that the equation of a circle is $(x-a)^2 + (y-b)^2$, and a sphere's equation similarly is $(x-a)^2 + (y-b)^2 + (z-c)^2$.
How do you derive the equation $\sqrt {a^2 - x^2 - y^2}$ from $(x-a)^2 + (y-b)^2 + (z-c)^2$?
I would appreciate it if you gave me some starting points for this proof, and not a full solution.
If the sphere is centered on the origin, you can ignore the constant terms you give as $a,~ b,~ c$ since they will all be 0. Then you are left with
$$x^2 + y^2 + z^2 = r^2 \\ z^2 = r^2-x^2-y^2 \\ z = \pm\sqrt{r^2-x^2-y^2}$$
And that person simply used $a$ instead of $r$.