Cylinder
A cylinder is a 3-D solid with two parallel circular bases connected by a curved lateral surface.
Formula
V = \pi r^2 h; \ SA = 2\pi r^2 + 2\pi r h
Definition
A cylinder is a 3-D shape with two parallel, congruent circular bases connected by a curved lateral surface perpendicular to the bases, like a soup can or paper towel roll; volume $V = \pi r^2 h$ and total surface area $SA = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)$, with the lateral surface, when unrolled, forming a rectangle of width $2\pi r$ and height $h$. Formally, a right circular cylinder is the solid of revolution obtained by rotating a rectangle about one of its sides, or equivalently the product of a disk $D(r)$ with an interval $[0,h]$; by integration, $$V = \int_0^h \pi r^2 \, dz = \pi r^2 h.$$
Example
A can with radius $4$ cm and height $10$ cm has volume $\pi \times 4^2 \times 10 = 160\pi = 502.7$ cm$^3$, with the label wrapped around the outside forming the lateral surface area. A cylindrical tank with diameter $2$ m and height $5$ m ($r = 1$ m) has $V = \pi(1)^2(5) = 5\pi = 15.71$ m$^3$ and $SA = 2\pi(1)(1+5) = 12\pi = 37.7$ m$^2$. The Pappus centroid theorem states that the surface area of a solid of revolution is $2\pi$ times the distance from the centroid of the generating curve to the axis of rotation, times the curve length; for a cylinder this gives $LSA = 2\pi r(h) = 2\pi r h$.
Key Insight
A cylinder is like a circle stretched into 3-D: every horizontal slice parallel to the base is a circle with the same radius, which is why volume = circle area $\times$ height, treating a cylinder as a prism with a circle as its base. For fixed volume, the cylinder that minimizes total surface area has $h = 2r$ (height equals diameter), which is why many cans are roughly as tall as they are wide. Pappus's theorem elegantly unifies the surface area and volume formulas for all solids of revolution.