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**Hint:**A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom.

The general formula for the total surface area of a cylinder is

\[2\pi rh + 2\pi {r^2}\]

Where the radius of the cylinder is r and altitude (height) is \[h\].

**Complete step-by-step answer:**

We need to derive the formula to find the total surface area of a cylinder.

Having radius \[r\] and altitude (height) \[h\], the surface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts:

The area of the top base: \[\pi {r^2}\]

The area of the bottom base: \[\pi {r^2}\]

The area of the side: \[2\pi rh\]

The area of the top and bottom bases is the same, and is called the base area, \[B\]. The area of the side is known as the lateral area, \[L\].

An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area)

\[L = 2\pi rh\]

The surface area of the solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore,

\[A = L + 2B = 2\pi rh + 2\pi {r^2}\]

Thus The general formula for the total surface area of a cylinder is \[2\pi rh + 2\pi {r^2}\] .

**Note:**A cylinder is one of the most basic curved geometric shapes, with the surface formed by the points at a fixed distance from a given line segment, known as the axis of the cylinder. The shape can be thought of as a circular prism. Both the surface and the solid shape created inside can be called a cylinder.

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