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The total surface area of a cylinder of radius r and height h is equivalent to 

A cylinder can be divided into three parts - two circles and a rectangle:

The total surface area of a cylinder is the sum of the areas of the circles and the rectangle.

The length of the rectangle is the height of the cylinder, i.e., h. The width of the rectangle is equivalent to the circumference of the circular bases, i.e., 2πr.

Thus, the area of the rectangle = length × width = h × 2πr = 2πrh = lateral surface area of a cylinder

Now, the two circular bases have a radius r of each.

Thus, area of each circular base = π × r² = πr²

the total surface area of the cylinder = area of the rectangle + area of two circles

= 2πrh + 2 × πr²

= 2πr (h + r)

Therefore, the total surface area of the cylinder = 2πr (h + r)