Cone Total Surface Area

Add the base disc to the curved side and you get the whole skin of a solid cone: $\pi r l + \pi r^2$, which factors neatly to $A = \pi r(r + l)$. The factored form is the one worth remembering, because it makes the structure obvious: one $\pi r$ multiplying a length that is part radius, part slant.

Whether you actually want the base is a judgement call the formula cannot make for you. Painting a conical spire? The base is the open bottom, so use the lateral area alone. Anodising a machined cone, sizing shrink-wrap for a pile of sand, or costing the sheet for a closed conical tank end? Then the base counts and this is your number.

Solving backwards for the radius is a quadratic, since $r$ appears squared in the base term. Both roots exist algebraically but only the positive one is a cone, which is why the solver returns $r = \tfrac{1}{2}\left(-l + \sqrt{l^2 + 4A/\pi}\right)$ and nothing else.