Cylinder Calculator

Computes geometric properties of a right circular cylinder from radius and height. Volume = πr²h. Lateral surface area = 2πrh. Total surface area = 2πr² + 2πrh.

Given any two of volume, radius, and height, the third can be solved. Output supports metric and imperial units with conversion to cubic feet, gallons, or liters where appropriate.

The right circular cylinder is one of the simplest 3D shapes with a closed-form expression for both volume and surface area, derived from elementary calculus. Volume is the integral of constant cross-sectional area πr² over height h, giving πr²h. Lateral (side wall) area unrolls into a rectangle of width 2πr (the circle's circumference) and height h, giving 2πrh. The two circular ends each contribute πr², for total surface area 2πr² + 2πrh = 2πr(r + h).

A worked example: a tank with radius 1 m and height 2 m has volume π × 1² × 2 ≈ 6.28 m³, or 6,283 liters, or 1,660 US gallons. Lateral area is 2π × 1 × 2 ≈ 12.57 m². Total area including the two end circles is 2π × 1 × (1 + 2) ≈ 18.85 m². For a sealed tank, the total area determines paint or insulation requirements; for an open-top container, lateral plus one end is the relevant figure.

The inverse problem (given volume, find dimensions) has infinitely many solutions: any radius can be paired with the appropriate height to hit a target volume. Engineering design often adds a constraint (minimum surface area for a given volume, fixed aspect ratio, or available material width) to fix one variable. The minimum-surface-area cylinder for a given volume has height = 2r (a 'square' cylinder where height equals diameter), which is why food cans approximate this ratio. Real cans deviate slightly to optimize for stacking, label area, and consumer perception of size.

Limitations are mostly about real-world deviations. Oblique cylinders (where the axis is not perpendicular to the base) have the same volume as right cylinders with the same base and perpendicular height (Cavalieri's principle), but the lateral surface area becomes a more complex elliptical calculation. Truncated cylinders (cut by a plane not parallel to the base) need conic-section formulas. Hollow cylinders (pipes) are computed as the difference between an outer and inner cylinder; the calculator handles this with separate inputs. Wall thickness for structural calculations involves the second moment of area, which is outside scope here.

For liquid storage, the volume calculation is exact, but real tanks add fittings, internal structure, and required headspace that reduce usable volume by 5-15% from the raw geometric figure.