What is the formula for the lateral surface area of a cylinder?

The formula for the lateral surface area of a cylinder is derived from the geometric properties of the shape. The lateral surface area refers to the area of the curved surface that connects the top and bottom circular bases of the cylinder.

This area can be visualized by imagining the cylinder being "unwrapped" into a rectangle. The height of the cylinder remains the same when unwrapped, and the width of the rectangle corresponds to the circumference of the base of the cylinder. The circumference of a circle is calculated using the formula (2\pi r), where (r) is the radius of the base.

Therefore, the lateral surface area can be calculated by multiplying the circumference of the base ((2\pi r)) by the height ((h)) of the cylinder. This leads to the formula:

[ \text{Lateral Surface Area} = \text{Circumference} \times \text{Height} = 2\pi r \times h ]

Thus, the correct answer captures this relationship accurately by incorporating both the height of the cylinder and the radius to determine the extensive area of the curved surface.