Lateral Surface Area
Lateral surface area is the total area of all the side faces of a solid, excluding the bases.
Formula
LSA = \text{perimeter of base} \times \text{height (prism)}
Definition
Lateral surface area is the area of just the sides of a 3-D shape, not counting the top or bottom, like the area of a label wrapped around a can. The lateral surface area of a prism is the sum of the areas of its rectangular side faces, $LSA = \text{perimeter of base} \times \text{height}$, which works for any prism regardless of the base shape; for a right cylinder, $LSA = 2\pi r h$ (the rectangle formed by unrolling the curved surface), and for a right cone, $LSA = \pi r l$ where $l$ is the slant height. Formally, for a right prism with base perimeter $P$ and height $h$, $LSA = Ph$; for an oblique prism, each face is a parallelogram and $LSA = Pl$ where $l$ is the lateral edge length, and for curved surfaces, LSA is the surface integral over the lateral portion only.
Example
A cereal box $8$ cm tall, $10$ cm wide, and $4$ cm deep has four side panels with areas $80$, $32$, $80$, and $32$ cm$^2$, giving lateral SA $= 224$ cm$^2$, not counting top or bottom. A triangular prism with base perimeter $18$ cm and height $10$ cm has $LSA = 18 \times 10 = 180$ cm$^2$, and a cone with radius $5$ cm and slant height $13$ cm has $LSA = \pi \times 5 \times 13 = 65\pi = 204.2$ cm$^2$. For a cone with half-angle $\alpha$ and height $h$, slant height $l = h/\cos\alpha$, radius $r = h\tan\alpha$, and $LSA = \pi r l = \pi h^2\tan\alpha/\cos\alpha$; as $\alpha \to 0$, the cone approaches a "needle" shape and LSA approaches $0$.
Key Insight
When you peel the label off a can and flatten it, you get a rectangle: that rectangle's area is the lateral surface area of the cylinder, a great way to visualize why $LSA = 2\pi r h$. In manufacturing, lateral surface area determines the material needed for cans, tubes, and pipes; the optimization problem of minimizing total material for a fixed volume leads to the result that the optimal cylinder has height equal to diameter, a classic result in applied calculus.