Cylinder Volume Calculator

Definition

The volume of a right circular cylinder is V = πr²h, where r is the radius of the circular base and h is the height measured perpendicularly between the bases. This calculator takes r and h directly. Units must stay consistent: if r and h are in inches, the result is in cubic inches.

A right circular cylinder has circular bases of equal radius 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. It is an exact geometric formula under the ideal-circle, constant-height assumptions — not an engineering approximation.

Historical and conceptual context

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

Conceptually the calculator does not infer material or pressure — it computes only the geometric interior volume of an ideal cylinder. If the cross-section is not a circle or the height is not constant, πr²h no longer applies. With diameter d = 2r the equivalent form is V = (πd²/4)h.

Formal properties

Input domain: r > 0, h > 0 (and finite floating-point numbers in practice). For r ≤ 0 or h ≤ 0 the geometric volume is not positive as a solid. Homogeneity: if every length is scaled by λ > 0, then r′ = λr, h′ = λh and V′ = λ³V, matching the dimension of volume.

At fixed r, V(h) = πr²·h is linear; at fixed h, V(r) grows like r². Relative sensitivity: ∂V/V = 2 ∂r/r + ∂h/h, so a relative radius error enters with weight 2. Base area A = πr², so V = A·h — the product of base area and height.

Full derivation from base area

Step 1. The area of a circle of radius r is A = πr². The constant π is the ratio of circumference to diameter.

Step 2. A right cylinder with circular base and height h has volume equal to base area times height: V = A·h = πr²h. This follows from Cavalieri’s principle: every cross-section parallel to the base has constant area A.

Step 3 (consistency with diameter). With d = 2r you get V = (πd²/4)h = πr²h. With circumference C = 2πr the same volume is C²h/(4π) — a separate page when you know C instead of r.

Step 4 (cone). A cone with the same r and h has volume (1/3)πr²h, one third of the cylinder. That is not this form’s output — only a geometric comparison.

Special cases

When r = 1 and h = 1, V = π ≈ 3.14 (the unit cylinder). When r = 5 and h = 10, V = 250π ≈ 785.40. When r = h = 4, V = 64π ≈ 201.06. The limit h → 0 yields V → 0 (a flattened cylinder). The limit r → 0 at fixed h also yields V → 0.

A square-cross-section prism needs a different formula (area = a², not πr²). A pipe of wall thickness δ needs a hollow-cylinder model: the difference of outer and inner volumes, not πr²h with the outer radius alone.

Numerical and computational notes

The calculator uses π in floating-point arithmetic and rounds the result to two decimal places, matching the worked examples on the page. Very large r²h may overflow the floating-point range; typical user dimensions are safe. The order π·r·r·h is algebraically equivalent to (πr²)h.

The code evaluates V = π · r · r · h. At r = 5 and h = 10: r² = 25, πr² ≈ 78.54, V = 250π ≈ 785.40. At r = 3 and h = 4: V = 36π ≈ 113.10.

Commas and periods in the fields are equivalent. The calculator does not model a cone, an oblique cylinder, or finite wall thickness. For an oblique cylinder the volume is still πr²h with perpendicular height (Cavalieri), but the slant is not h — this form assumes a right cylinder.

Related ideas and sources

Related models on this site: circle area (πr²), cylinder volume from circumference (C²h/(4π)), cylinder lateral area (2πrh), cone volume ((1/3)πr²h), and sphere volume ((4/3)πr³). This formula computes only the volume of a solid right circular cylinder from r and h.

A cone with the same r and h has volume (1/3)πr²h — one third of this cylinder; a separate calculator applies the factor 1/3. Lateral area 2πrh and total surface area are different questions. Do not confuse height h with the slant length along the lateral surface.

Sources: Archimedes, On the Sphere and Cylinder (Heath); Euclid, Elements, Book XII; NIST DLMF (π); MathWorld — Cylinder; OpenStax — volume of cylinders; Wikipedia — Cylinder (geometry). The calculator is an educational tool and does not replace engineering standards for pressure vessels.