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Consider a sphere $Gamma$ centered around the origin and of radius $R$.
Consider a circle $gamma_0$ of radius $r_0 < R$ on the sphere $Gamma$. The circle is centered around the $z$-axis and is in a plane parallel to the $(x,y)$ plane, at $z_0 > 0$.
Let $A_0$ be the surface area on the sphere $Gamma$ bounded by the circle $gamma_0$ and with $z ge z_0$. Let $V_0$ be the volume of the ‘ice cream’ cone consisting of all points on line segments joining the origin to points on $A_0$. Let $A_1$ be the surface area of the side of the cone $V_0$.

Provide expressions for the area element $dS$ of the ‘ice cream’ cone volume $V_0$ over its surface, $S_0 = A_0 + A_1$.

I can quite easily find the area element corresponding to $A_0$ as this is simply $$r^2sintheta,dphi ,dtheta ,bf{r}$$ where $bf{r}$ is a unit vector. But I am struggling to find out the expression for $A_1$ as the side of the cone doesn't seem logical to use spherical polars to find the surface area.

Any help would be really appreciated. Thanks

I don't know if this is exactly what you're looking for, but if I have a cone whose tip is located at the origin and whose sides slope at an angle $phi$ away from the positive $z$-axis:

Then you can parametrize this surface using polar coordinates like so:
$$ vec S(r,theta) = (r cos theta, r sin theta, r cot phi) $$
The area element is then
$$ dS = left|frac{dvec S}{dr} times frac{dvec S}{dtheta} right| , dr, dtheta $$
which yields if you go thru all the trouble:
$$ dS = r csc phi, dr, dtheta $$