Which of the following formulas represents the lateral surface area of a cone?

The lateral surface area of a cone can be understood as the area of the curved surface that connects the base of the cone to its apex. The formula that accurately represents this quantity is derived from the geometry of the shape.

The correct formula involves multiplying the radius of the base of the cone by the slant height. This is because the lateral area can be visualized as a sector of a circle when the cone is "unwrapped" into a two-dimensional shape. The slant height serves as the radius of this sector, while the arc length corresponds to the circumference of the base of the cone (which is calculated as 2 times pi times the radius).

Since the lateral surface area is essentially the area of this sector of the circle, it can also be expressed as the product of the circumference of the base of the cone (which is pi times the diameter or 2 times pi times the radius) and the height of the cone as a proportion of the slant height. The correct formula simplifies to pi times the radius times the slant height, effectively linking the dimensions of the cone to its lateral surface area.

Understanding this formula is critical for problems that involve calculating the surface area of conical shapes in various applications, allowing for practical and theoretical