Volume of a Cylinder
The volume of a cylinder equals $\pi$ times the radius squared times the height.
Formula
V = \pi r^2 h
Definition
To find the volume of a cylinder, find the area of the circular base ($\pi r^2$) and multiply by the height: $V = \pi r^2 h$, equal to the base area times height, consistent with the general prism formula $V = Bh$ since a cylinder is a prism with a circular base. Doubling the radius quadruples the volume (since $r$ is squared), while doubling the height only doubles it. Derived by integrating circular cross-sections, $$V = \int_0^h \pi r^2 \, dz = \pi r^2 h,$$ or equivalently by the shell method, $$V = \int_0^r 2\pi t h \, dt = \pi r^2 h,$$ both methods agreeing, illustrating that volume is independent of the integration approach.
Example
A can with radius $3$ cm and height $10$ cm has $V = \pi \times 9 \times 10 = 90\pi = 282.7$ cm$^3$. A cylindrical pool $6$ m in diameter ($r=3$ m) and $1.5$ m deep has $V = \pi(9)(1.5) = 13.5\pi = 42.4$ m$^3 = 42{,}412$ liters. Optimizing a cylinder of fixed surface area $SA = 2\pi r^2 + 2\pi r h$ for maximum volume shows the maximum occurs at $h = 2r$ (height equals diameter), the "ideal can" problem in calculus optimization.
Key Insight
The cylinder volume is exactly $3$ times the cone volume with the same base and height (since $V_{cone} = (1/3)\pi r^2 h$), which is why three ice cream scoops from a cone-shaped server equal one cylindrical cup. The optimal can problem shows that minimizing material for a fixed volume and maximizing volume for fixed material both give $h = 2r$, though many real cans deviate from this because of manufacturing costs, stacking efficiency, and lid-to-body material ratios.