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Hollow Cylinder Calculator

Volume equals pi times the quantity outer radius squared minus inner radius squared, times height

Solution

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Calculate Hollow Cylinder Volume from R, r, and Height

Use this form when you know the outer radius, inner radius, and height of a pipe, tube, or hollow cylinder and need the volume of the wall material — the steel in a pipe, the insulation around a coaxial cable, or the cardboard in a paper tube.

V = π (R² − r²) h

Calculate Total Surface Area of a Hollow Cylinder

Use this form when you need the area of both annular end caps plus the outer and inner lateral surfaces. Common for painting, coating, plating, or galvanizing the entire wetted area of a tube.

S = 2π [(R² − r²) + (R + r) h]

Calculate Hollow Cylinder Outer Radius (R)

Use this rearrangement when you know the required wall volume, the inner bore radius (fixed by what must pass through the pipe), and the available height — and need the minimum outer radius that supplies that wall volume.

R = √(V / (π h) + r²)

Calculate Hollow Cylinder Inner Radius (r)

Use this rearrangement when the outer radius is fixed by stock size, the height is known, and you need to size the bore. Common when boring out a solid blank to leave a target amount of wall material.

r = √(R² − V / (π h))

Calculate Hollow Cylinder Height

Use this rearrangement when the cross-section (R, r) is fixed by an existing pipe spec and you need the length that contains a target wall volume — for example, cutting a tube to match a required mass.

h = V / (π (R² − r²))

How It Works

This hollow cylinder calculator uses V = π (R² − r²) h plus the surface-area formula S = 2π [(R² − r²) + (R + r) h] to solve for any of volume, total surface area, outer radius, inner radius, or height. Pick the unknown with the solve-for toggle, enter the remaining three values with any supported length, area, or volume units, and the calculator converts to SI internally before computing all related quantities — including the annular cross-section area Aₓₛ = π (R² − r²), wall thickness t = R − r, outer diameter d₀ = 2R, and inner diameter dᵢ = 2r — so you can size pipes, tubes, sleeves, coaxial cables, and pressure-vessel shells from a single page.

Example Problem

A steel pipe segment has an outer radius of 5 cm, an inner radius of 3 cm, and a length of 10 cm. What volume of steel does the wall contain?

  1. Identify the three measured dimensions: outer radius R = 5 cm, inner radius r = 3 cm, and length (height) h = 10 cm.
  2. Choose the unknown: we want the wall volume, so use V = π (R² − r²) h.
  3. Square each radius: R² = 25 cm², r² = 9 cm². The annular cross-section area is Aₓₛ = π (25 − 9) = 16π ≈ 50.27 cm².
  4. Multiply by the height: Aₓₛ · h = 16π · 10 = 160π cm³.
  5. Evaluate: V = 160π ≈ 502.65 cm³.
  6. Cross-check the wall thickness: t = R − r = 5 − 3 = 2 cm, and the outer / inner diameters are d₀ = 10 cm and dᵢ = 6 cm — useful for matching the pipe to a real-world spec table.

Round trip check: from V ≈ 502.65 cm³, r = 3 cm, and h = 10 cm, the inverse formula R = √(V / (π h) + r²) recovers R = 5 cm, confirming the forward calculation.

When to Use Each Variable

  • Solve for Volume — when R, r, and h are known and you need the volume of wall material — pipe steel, insulation jacket, sleeve plastic — for mass, weight, or material-cost calculations.
  • Solve for Total Surface Area — when you need the full wetted area of a pipe or tube — both end-cap annuli plus the outer and inner lateral surfaces — for coating, plating, or heat-transfer area estimates.
  • Solve for Outer Radius — when the bore size and length are fixed and you need the minimum outer radius that provides a target wall volume.
  • Solve for Inner Radius — when the outer dimensions are fixed by stock material and you need to size the bore to leave a target amount of wall material.
  • Solve for Height — when the pipe cross-section (R, r) matches an existing spec and you need the cut length that holds a target wall volume or mass.

Key Concepts

A hollow circular cylinder — a pipe, tube, or sleeve — is the solid that remains when a smaller coaxial cylinder of radius r is removed from a larger cylinder of radius R, with both cylinders sharing the same height h. Its cross-section perpendicular to the axis is an annulus (ring) with area π (R² − r²), and its wall volume is that cross-section multiplied by the height. The total surface area sums four pieces: two annular end caps, the outer lateral surface (a tall rectangle of width 2π R and height h when unrolled), and the inner lateral surface (a rectangle of width 2π r and height h). Wall thickness t = R − r and the outer / inner diameters d₀ = 2R, dᵢ = 2r are the parameters most often quoted on real-world pipe and tube spec sheets — divide by two to get the radii the formulas need.

Applications

  • Piping and pressure vessels: compute pipe wall volume for material mass, weight, or hoop-stress inputs from an outer diameter, inner diameter (or wall thickness), and length
  • Coaxial cables and conductors: find the cross-section area and volume of the insulating dielectric or the conductive shield between the inner core and the outer braid
  • Bearings, bushings, and sleeves: compute the volume of material in a cylindrical sleeve being bored from a solid blank, or the surface area for plating and coating
  • Heat exchangers and double-pipe geometry: the annular flow passage between an inner and outer pipe shares the same hollow-cylinder cross-section formulas
  • Paper tubes, mailing tubes, and packaging: estimate cardboard or composite mass per tube for shipping cost and inventory planning

Common Mistakes

  • Confusing the wall volume V = π (R² − r²) h with the bore volume π r² h (the fluid capacity inside the pipe) — this calculator returns the wall material volume, not the hollow interior
  • Using diameters instead of radii — pipe specs usually quote outer diameter and inner diameter or wall thickness; divide by two before plugging into the formulas, or convert d₀, dᵢ to R = d₀/2 and r = dᵢ/2
  • Computing wall volume from outer volume minus inner volume but mixing length units (R in mm, h in m) — convert everything to a single length unit first or let the calculator's unit dropdowns handle it
  • Forgetting the inner lateral surface when computing total surface area — a tube has an inside as well as an outside, so the total surface area is larger than the outer-surface-only value
  • Treating wall thickness t and inner radius r as interchangeable — they are related by r = R − t and d_i = d_o − 2t, not by r = t

Frequently Asked Questions

How do you calculate the volume of a hollow cylinder?

Use V = π (R² − r²) h, where R is the outer radius, r is the inner radius, and h is the height (length) of the pipe or tube. Square each radius, subtract, multiply by the height, and multiply by π. For example, a pipe with R = 5 cm, r = 3 cm, and h = 10 cm contains V = π · 16 · 10 = 160π ≈ 502.65 cm³ of wall material.

What is the formula for hollow cylinder surface area?

Total surface area is S = 2π [(R² − r²) + (R + r) h], which equals the two annular end caps (2π (R² − r²)) plus the outer lateral surface (2π R h) plus the inner lateral surface (2π r h). For R = 5, r = 3, h = 10 the total is 192π ≈ 603.19 cm².

What is the difference between a hollow cylinder and an annulus?

An annulus is the two-dimensional ring shape between two concentric circles, with area π (R² − r²). A hollow cylinder is the three-dimensional pipe-shaped solid whose cross-section is that annulus — extruding an annulus a distance h along its axis gives a hollow cylinder. The annulus is the cross-section; the hollow cylinder is the full solid.

How do you calculate the volume of a pipe wall?

A pipe wall is a hollow cylinder, so its volume is V = π (R² − r²) h. If you only have the outer diameter d₀ and wall thickness t, compute R = d₀ / 2 and r = R − t first. The wall volume multiplied by the material density gives the pipe's mass — useful for shipping, hanging, and stress calculations.

What is the cross-section area of a hollow cylinder?

The cross-section perpendicular to the axis is an annulus with area Aₓₛ = π (R² − r²). For a pipe with R = 5 cm and r = 3 cm, Aₓₛ = π · 16 ≈ 50.27 cm². This is the area carrying axial load in a column, and the area through which heat conducts axially.

How do you find wall thickness from inner and outer radius?

Wall thickness is simply t = R − r — the difference between the outer and inner radii. Equivalently, from diameters, t = (d₀ − dᵢ) / 2. The calculator displays t alongside every solve result so you can match it against a pipe schedule (for example, Schedule 40 vs Schedule 80) directly.

Can I use this calculator for a solid cylinder?

Yes — set the inner radius r = 0 and the formulas reduce to the solid cylinder versions: V = π R² h and S = 2π R (R + h). If you only ever need solid-cylinder math, the dedicated cylinder calculator may be a cleaner fit.

How is hollow cylinder volume used in engineering?

Wall volume multiplied by material density gives the mass of a pipe, sleeve, or bushing — the basis for shipping weight, structural dead load, and hoop-stress / longitudinal-stress calculations. The cross-section area Aₓₛ shows up in axial stress (σ = F / Aₓₛ) and in moment-of-inertia formulas for bending, where the hollow cross-section is much stiffer than a solid bar of equal mass.

Reference: Weisstein, Eric W. "Cylindrical Shell." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/CylindricalShell.html

Worked Examples

Steel Pipe

How do you calculate the wall volume of a steel pipe?

A 1-meter length of steel pipe has an outer diameter of 100 mm (R = 50 mm = 0.05 m) and a wall thickness of 10 mm (r = 40 mm = 0.04 m). Use V = π (R² − r²) h to find the volume of steel — the basis for the pipe's mass and shipping weight.

  • Knowns: R = 0.05 m, r = 0.04 m, h = 1 m
  • Cross-section area: Aₓₛ = π (R² − r²) = π (0.0025 − 0.0016) = π · 0.0009 ≈ 2.827 × 10⁻³ m²
  • Wall volume: V = Aₓₛ · h ≈ 2.827 × 10⁻³ m³
  • Approximate steel mass at 7,850 kg/m³: m ≈ 7,850 · 2.827 × 10⁻³ ≈ 22.2 kg per meter

Wall volume ≈ 2.83 × 10⁻³ m³ per meter of pipe; mass ≈ 22.2 kg/m at typical steel density

Real pipe schedules (Schedule 40, Schedule 80) specify wall thickness directly. From t and the outer diameter d₀, compute r = d₀/2 − t before using the formula.

Paper Tube

How much cardboard is in a mailing tube?

A poster-shipping tube is 60 cm long, with an outer radius of 4 cm and a wall thickness of 3 mm — so the inner radius is r = 4 cm − 0.3 cm = 3.7 cm. Find the cardboard cross-section and volume.

  • Knowns: R = 4 cm, r = 3.7 cm, h = 60 cm
  • Cross-section area: Aₓₛ = π (16 − 13.69) = π · 2.31 ≈ 7.257 cm²
  • Cardboard volume: V = Aₓₛ · h ≈ 7.257 · 60 ≈ 435.4 cm³
  • At a typical cardboard density of ~0.7 g/cm³, the tube mass is m ≈ 0.7 · 435.4 ≈ 305 g

Cardboard volume ≈ 435 cm³, mass ≈ 0.30 kg per tube

Paper tubes are often quoted by inner diameter and wall thickness. Convert to (R, r) before computing — and remember caps or end-plugs are extra material not included in V.

Coaxial Cable Insulation

What is the dielectric volume in a coaxial cable?

A 50 Ω coaxial cable has an inner conductor radius of 0.5 mm and a dielectric outer radius of 1.65 mm (the outer braid sits at this radius). For a 10-meter length, what volume of dielectric (PTFE / polyethylene) fills the cable?

  • Knowns: R = 1.65 mm, r = 0.5 mm, h = 10 m = 10,000 mm
  • Cross-section area: Aₓₛ = π (R² − r²) = π (2.7225 − 0.25) = π · 2.4725 ≈ 7.768 mm²
  • Dielectric volume: V = Aₓₛ · h ≈ 7.768 · 10,000 ≈ 77,680 mm³ ≈ 77.68 cm³ ≈ 7.77 × 10⁻⁵ m³
  • At polyethylene density ~0.93 g/cm³, mass ≈ 0.93 · 77.68 ≈ 72.2 g per 10 m

Dielectric volume ≈ 77.7 cm³ per 10 m of cable; mass ≈ 72 g at polyethylene density

The 50 Ω impedance comes from the ratio R/r ≈ 3.3 combined with the dielectric constant — the hollow-cylinder geometry alone fixes capacitance and inductance per unit length.

Hollow Cylinder Formulas

All hollow-cylinder (pipe, tube, sleeve) properties are determined by three dimensions: the outer radius R, the inner radius r, and the height (length) h. From those, the wall volume, total surface area, annular cross-section area, wall thickness, and both diameters all follow:

Volume equals pi times the quantity outer radius squared minus inner radius squared, times height
Total surface area equals two pi times the quantity outer radius squared minus inner radius squared, plus the quantity outer radius plus inner radius, times height
Outer radius equals the square root of volume divided by pi times height, plus inner radius squared
Inner radius equals the square root of outer radius squared minus volume divided by pi times height
Height equals volume divided by pi times the quantity outer radius squared minus inner radius squared
Hollow cylinder (pipe) with labeled outer radius R, inner radius r, and height hRrh

Where:

  • V — wall (material) volume: the volume of pipe steel, sleeve plastic, or cable dielectric — not the bore volume π r² h
  • S — total surface area: two annular end caps + outer lateral surface + inner lateral surface
  • Aₓₛ — annular cross-section area, π (R² − r²); the load-bearing area in axial-stress problems
  • R — outer radius (m, cm, mm, in, ft, yd)
  • r — inner radius (bore radius); must satisfy 0 ≤ r < R
  • h — height (axial length) between the two end caps
  • t — wall thickness, t = R − r
  • d₀ — outer diameter, d₀ = 2R (commonly quoted on pipe spec sheets)
  • dᵢ — inner diameter (bore), dᵢ = 2r

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