Cylinder
Geometry & MeasurementA 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 circular ends (bases) that are the same size and a curved surface connecting them. A soup can and a paper towel roll are cylinders.
Example
A can with radius $4$ cm and height $10$ cm: Volume $= \pi \times 4^2 \times 10 = 160\pi = 502.7$ cm$^3$. The label wrapped around the outside is the lateral surface area.
Key Insight
A cylinder is like a circle stretched into 3-D. Every horizontal slice through a cylinder parallel to the base is a circle with the same radius. This is why volume = circle area $\times$ height.
Definition
A right circular cylinder has two parallel, congruent circular bases and a lateral surface perpendicular to the bases. Volume $V = \pi r^2 h$; Total surface area $SA = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)$. The lateral surface, when unrolled, forms a rectangle of width $2\pi r$ and height $h$.
Example
A cylindrical tank with diameter $2$ m and height $5$ m: $r = 1$ m. $V = \pi(1)^2(5) = 5\pi = 15.71$ m$^3$. $SA = 2\pi(1)(1+5) = 12\pi = 37.7$ m$^2$.
Key Insight
For fixed volume, the cylinder that minimizes total surface area has $h = 2r$ (height equals diameter). This is why many cans are roughly as tall as they are wide: it is the most material-efficient shape.
Definition
A right circular cylinder is the solid of revolution obtained by rotating a rectangle about one of its sides. Alternatively, it is the product of a disk $D(r)$ with an interval $[0,h]$. Volume by integration: $$V = \int_0^h \pi r^2 \, dz = \pi r^2 h.$$ The surface area is derived using the surface area integral for surfaces of revolution.
Example
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: centroid of the lateral rectangle side at distance $r$, so $LSA = 2\pi r(h) = 2\pi r h$.
Key Insight
Pappus's theorem elegantly unifies the surface area and volume formulas for all solids of revolution. It connects the geometry of 2-D centroids to the 3-D measurement of volumes and surface areas, enabling rapid computation without integration.