What is the equation of a sphere of radius R centred at the origin in cylindrical coordinates?

Question

I said that $r = R, -R \leq z \leq R$, and $0 \leq \theta \leq 2\pi$.

Saying that $r = R$ is incorrect, however, but I don't understand why because clearly, at all points of the sphere the radius is $R$.

Answer 1

In rectangular coordinates the equation is $$x^2+y^2+z^2=R^2$$ For cylindrical coordinates, z stays as is, and $r^2=x^2+y^2$. So you have $$r^2+z^2=R^2$$ Remember that for cylindrical coordinates r is a distance to the z axis, not the distance to the origin.