terican

Last verified · v1.0

Calculator · math

Hollow Cylinder Volume Calculator

Compute the volume of a hollow cylinder or pipe using V = π·h·(R²−r²). Enter outer radius, inner radius, and height for instant results in any unit system.

FreeInstantNo signupOpen source

Inputs

Hollow Cylinder Volume

—

Explain my result

0/3 free

Get a plain-English breakdown of your result with practical next steps.

Hollow Cylinder Volume—cubic units

The formula

How the
result is
computed.

Hollow Cylinder Volume Calculator — Formula and Methodology

A hollow cylinder — also called a cylindrical shell or annular cylinder — describes any solid bounded by two concentric cylindrical surfaces of different radii and a fixed height. Everyday examples include steel pipes, PVC tubes, cardboard rolls, concrete pillars, and engine sleeves. This calculator applies the standard volume formula to deliver fast, accurate results.

The Core Formula: V = π · h · (R² − r²)

The formula derives directly from the subtraction principle: the hollow cylinder occupies the space of the outer solid cylinder minus the inner cylindrical void.

  • Outer cylinder volume: Vouter = π · R² · h
  • Inner void volume: Vinner = π · r² · h
  • Hollow cylinder volume: V = π · R² · h − π · r² · h = π · h · (R² − r²)

Variable Definitions

  • R (outer radius) — distance from the central axis to the outer surface. Always the larger of the two radii.
  • r (inner radius) — distance from the central axis to the inner wall, forming the hollow core. Must satisfy r < R.
  • h (height) — the perpendicular distance between the two circular bases, sometimes called the length for pipes and tubes.
  • π — the mathematical constant pi, approximately 3.14159265358979.

Wall Thickness and Geometric Constraints

The wall thickness of a hollow cylinder is computed as T = R − r, representing the radial distance of solid material between the inner and outer surfaces. This geometric relationship is fundamental to engineering design. The constraint r < R ensures strictly positive wall thickness; zero or negative thickness contradicts the physical definition of a hollow cylinder with material substance. Engineers leverage wall thickness to optimize structural strength, weight efficiency, and material cost — thicker walls increase volume and strength capacity, while thinner walls reduce material usage. The interplay between outer radius, inner radius, and wall thickness determines the cylinder's load-bearing capacity and practical viability.

Calculus Derivation via the Shell Method

The shell method in integral calculus confirms this result independently. A thin rectangular strip at distance x from the central axis, with height h and infinitesimal width dx, sweeps out a cylindrical shell with volume dV = 2π · x · h · dx when rotated around the axis. Integrating from r to R gives:

V = ∫ from r to R of 2π · x · h dx = π · h · [x²] evaluated from r to R = π · h · (R² − r²)

This derivation is covered in depth by Lamar University's Calculus I — Volumes of Solids of Revolution: Method of Cylinders and in the Rochester Institute of Technology's Volumes by Integration reference handout.

Worked Example 1 — Steel Pipe

A steel pipe has an outer radius R = 5 cm, inner radius r = 4 cm, and length h = 100 cm. Applying the formula:

  • R² = 25 cm², r² = 16 cm², R² − r² = 9 cm²
  • V = π × 100 × 9 = 900π ≈ 2,827.43 cm³
  • Wall thickness = R − r = 1 cm; material volume confirms the 1 cm shell.

Worked Example 2 — PVC Tube

A PVC tube has an outer radius R = 3 inches, inner radius r = 2.5 inches, and length h = 24 inches.

  • R² = 9 in², r² = 6.25 in², R² − r² = 2.75 in²
  • V = π × 24 × 2.75 = 66π ≈ 207.35 in³

Unit System Reference

The calculator supports millimeters (mm), centimeters (cm), meters (m), inches (in), and feet (ft). Output volume units follow the input system automatically — cm inputs produce cm³ output, inch inputs produce in³ output. Key conversions: 1 in³ ≈ 16.387 cm³; 1 cm³ = 0.001 liters; 1 ft³ ≈ 28,316.85 cm³.

Practical Applications

  • Plumbing and piping: Determine pipe material volume to estimate weight, raw material cost, and shipping expenses for steel, copper, or CPVC pipe runs in residential and industrial systems.
  • Manufacturing: Calculate shell volume for bushings, bearings, seals, and annular machine components during mass production planning, stress analysis, and cost estimation.
  • Civil engineering: Estimate concrete volume for hollow columns, reinforced piles, tunnel lining segments, and structural shafts that require precise material quantification.
  • Education: Demonstrate the volume subtraction principle visually in geometry and integral calculus coursework, reinforcing conceptual understanding through interactive computation.

Reference

Frequently asked questions

What is the formula for the volume of a hollow cylinder?
The volume of a hollow cylinder is V = π·h·(R² − r²), where R is the outer radius, r is the inner radius, and h is the height. The formula subtracts the inner cylindrical void volume (π·r²·h) from the outer solid cylinder volume (π·R²·h), leaving only the material volume of the cylindrical wall. For example, a pipe with R = 5 cm, r = 4 cm, and h = 100 cm has a material volume of approximately 2,827.43 cm³.
What is the difference between a hollow cylinder and a solid cylinder in terms of volume?
A solid cylinder has no central hole, so its volume formula is simply V = π·R²·h. A hollow cylinder has a cylindrical void of inner radius r through its center, reducing its volume to V = π·h·(R² − r²). For a concrete column with R = 20 cm and h = 300 cm, the solid version holds 376,991 cm³ of material, while a hollow version with r = 15 cm holds only 220,907 cm³ — a 41.4% reduction in material for a hollow core.
How do I find the volume of a pipe using this hollow cylinder calculator?
To find pipe volume, enter the outer radius R (half the outer diameter), the inner radius r (half the inner diameter or bore diameter), and the pipe length as the height h. For a standard schedule-40 steel pipe with 2-inch nominal size, the outer diameter is 2.375 in and inner diameter is 2.067 in, giving R = 1.1875 in and r = 1.0335 in. Enter any desired pipe length for h. The calculator instantly returns V = π·h·(R² − r²) in cubic inches.
What happens if the inner radius is greater than or equal to the outer radius?
The inner radius must always be strictly less than the outer radius (r < R). If r equals R, the wall thickness becomes zero and the formula yields V = 0, representing no solid material. If r exceeds R, the result is mathematically negative and physically meaningless. The hollow cylinder volume calculator enforces the r < R constraint and alerts users to invalid inputs before any calculation proceeds, preventing nonsensical results.
What unit systems does the hollow cylinder volume calculator support?
The calculator supports millimeters (mm), centimeters (cm), meters (m), inches (in), and feet (ft) for all three input variables. The output volume unit matches the selected system — entering measurements in centimeters produces a result in cubic centimeters (cm³), while inches produce cubic inches (in³). To convert cm³ to liters, divide by 1,000. To convert in³ to US gallons, divide by 231. To convert ft³ to liters, multiply by 28.3168.
How is the hollow cylinder volume formula proven using integral calculus?
The cylindrical shell method provides a rigorous calculus proof. Rotating a thin strip at distance x from the axis with height h through 360° generates a shell of volume dV = 2π·x·h·dx. Integrating over the wall thickness from inner radius r to outer radius R gives V = 2π·h · ∫ from r to R of x dx = 2π·h · [(x²/2)] from r to R = π·h·(R² − r²). This approach, detailed in Lamar University's calculus resources and the RIT integration handout, confirms the algebraic subtraction formula.