Cylinder volume from circumference

Definition

The volume of a right circular cylinder is classically V = πr²h, where r is the base radius and h is the height measured perpendicularly between the bases. This calculator takes the base circumference C and the height h instead of r. From C = 2πr one obtains r = C/(2π). Substituting yields V = π · (C/(2π))² · h = C²h/(4π). Units must be consistent: if C and h are in metres, the result is in cubic metres.

A right circular cylinder has congruent circular bases and generators perpendicular to the base planes. The formula V = πr²h is the volume of the solid of revolution obtained by rotating a rectangle about one side. The form C²h/(4π) does not change the geometry — it only removes the need to supply the radius when the circumference is known (a tape around a pipe, a hoop, or a rolled sheet).

Historical and conceptual context

Cylinder volume belongs to the oldest results in solid geometry. In On the Sphere and Cylinder Archimedes proved that a sphere has two-thirds the volume of its circumscribed cylinder and two-thirds the total surface of that cylinder; those proportions already assume V_cylinder = πr²h. Euclid’s Elements, Book XII, treats cylinders and cones by exhaustion, comparing solids of equal height.

A circumference-based formula is convenient in engineering and school practice: one often measures the girth of a pipe or tank rather than an internal diameter. Conceptually the calculator computes only the geometric interior volume of an ideal cylinder. If the cross-section is not a circle or the height is not constant, C²h/(4π) does not apply.

Formal properties

Input domain: C > 0, h > 0 (and finite floating-point values in practice). For C ≤ 0 or h ≤ 0 there is no positive solid volume in this model. Homogeneity: V scales quadratically in C and linearly in h. If all lengths are multiplied by λ > 0, then C′ = λC, h′ = λh and V′ = λ³V, matching the dimension of volume.

Equivalent forms: V = πr²h = (πd²/4)h = C²h/(4π), where d = 2r = C/π. Base area A = πr² = C²/(4π), so V = A·h. Mean curvature and other differential quantities of the surface do not enter this model.

For fixed C, V(h) = (C²/(4π))·h is linear; for fixed h, V(C) grows like C². Relative sensitivity: ∂V/V = 2 ∂C/C + ∂h/h, so a relative error in circumference enters with weight 2.

Full derivation from circumference

Step 1. The circumference of a circle of radius r is C = 2πr. Solving for r gives r = C/(2π).

Step 2. Base area: A = πr². Substituting r yields A = π · C²/(4π²) = C²/(4π).

Step 3. Volume of a right cylinder of height h: V = A·h = C²h/(4π). Equivalently V = π(C/(2π))²h = C²h/(4π).

Step 4 (consistency with diameter). With d = C/π one has V = (πd²/4)h = (π/4)·(C/π)²·h = C²h/(4π). The three routes (r, d, C) agree.

Special cases

If C = 2π (length units), then r = 1 and V = πh. If C = π, then d = 1 and V = (π/4)h. For a pipe of circumference about 62.8 cm ≈ 2π·10 cm and height 50 cm, V ≈ C²h/(4π) ≈ 15 700 cm³ ≈ 15.7 L — assuming an interior circumference; an exterior tape overestimates volume by the wall thickness.

As h → 0, V → 0. As C → 0 with fixed h, V → 0. A cylinder with square cross-section needs a different formula (area a², not C²/(4π)).

Numerical and computational notes

The calculator uses π in floating-point arithmetic. Very large C²h may overflow the floating-point range; ordinary workshop sizes are safe. Evaluating C·C·h/(4·π) is algebraically equivalent to the radius form and avoids an intermediate division by 2π before squaring when C is already given.

A comma or a period in the fields is accepted equivalently. The calculator does not model cones, oblique cylinders, or finite wall thickness.

In laboratory and school practice one must distinguish interior volume from the solid implied by an exterior girth: an exterior tape measures C_ext = 2π(R+δ) for wall thickness δ, while liquid fills radius R. Feeding C_ext into C²h/(4π) overestimates the capacity. The correction uses an interior circumference or C_int = C_ext − 2πδ for constant thickness. The formula ignores meniscus, welds, and dished tank ends.

Comparison with an oblique cylinder: by Cavalieri’s principle an oblique circular cylinder of perpendicular height h still has volume πr²h, but the slant generator is not equal to h. This calculator assumes a right cylinder and takes h as perpendicular height. The constant π in the denominator 4π comes only from circle geometry, not from an engineering approximation.

Dimensional check: if C and h have dimension L, then C²h/π has dimension L³, matching volume. Replacing π by a rational approximation (22/7 or 3.14) changes only the numerical constant, not the algebraic dependence on C and h. For precise work the calculator keeps π rather than a truncated decimal.

Related concepts and authorities

Related tools on this site: circle circumference (C = 2πr), cylinder volume from radius, circle area, and volume-unit converters. This formula does not compute lateral area 2πrh or total surface area.

Authorities: Archimedes, On the Sphere and Cylinder (Heath); Euclid, Elements, Book XII; NIST DLMF (π); MathWorld — Cylinder; Wikipedia — Cylinder (geometry). The calculator is educational and does not replace pressure-vessel design codes.