Surface Area of Cone
The total surface area of a cone consists of two parts:

The curved surface area

The flat surface area (of the base)
The flat surface area is simply \(\pi {r^2}\), since the base is circular. What is the curved surface area? To determine that, visualize what would happen if we cut open the cone and roll it out:
We will obtain a circular sector. It would be a good idea to try this activity. Cut out a circular sector from a piece of paper. Then, fold / roll the paper so that the two straight edges align with each other, and you will get a cone.
Note that the radius R of the circular sector is the slant height of the cone. Denote the slant height by \(l\). Observe that \(l= \sqrt {{r^2} + {h^2}} \). Also, the arc length L of the circular sector is the circumference of the cone’s base, which is \(2\pi r\). Now, the area of the circular sector is
\[\begin{align}&A = \frac{1}{{2\pi }} \times \frac{L}{R} \times \pi {R^2}\,\,\,\left( {{\rm{why}}?} \right)\\&\,\,\,\,\, = \frac{{LR}}{2} = \frac{{\left( {2\pi r} \right)\left( l \right)}}{2}\\&\,\,\,\,\, = \pi rl\end{align}\]
This is also the curved surface area of the cone!
To summarize:

Flat SA= \(\pi {r^2}\)

Curved SA = \(\pi rl = \pi r\sqrt {{r^2} + {h^2}} \)

Total SA = \(\pi {r^2} + \pi rl = \pi r\left( {r + l} \right)\)