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hollow cylinder

October 3, 2025
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The hollow cylinder is one of the most commonly used geometric shapes in engineering, architecture, and industry. This three-dimensional figure, consisting of two concentric cylinders with different radii, is widely   used in everyday life and in industry    . This article discusses in detail the definition of the hollow cylinder, its mathematical properties, calculation formulas, and applications.

Definition of a hollow cylinder

A hollow cylinder is a three-dimensional volume formed by the difference between two right circular cylinders with identical axes and heights. In other words, if you separate a smaller cylinder from a larger cylinder with the same height and axis, you get a hollow cylinder.

Components of a hollow cylinder:

  1. Inner radius (r)     : The inner radius of the cylinder.

  2. Outer radius (R)     : The outer radius of the cylinder.

  3. Height (h)     : Distance between the bottom of the cylinder.

  4. Wall thickness (t)     : the difference between outer radius and inner radius (t = R – r)

Geometric properties of hollow cylinders

1. Surface

The total surface of a hollow cylinder consists of three parts:

  • Outer surface area     : 2πRh

  • Inner surface area     : 2πrh

  • The area of ​​the lower and upper ring     is 2π(R² – r²)

Formula for the total area     :
A = 2πRh + 2πrh + 2π(R² – r²) = 2πh(R + r) + 2π(R² – r²)

2. Volume of a hollow cylinder

The volume of a hollow cylinder is equal to the difference between the volumes of the outer and inner cylinders:

Volume formula     :
V = πR²h – πr²h = πh(R² – r²)

Using the sum of squares, this formula can also be written as follows:
V = πh(R – r)(R + r)

3. Moment of inertia

In technical calculations, the moment of inertia of a hollow cylinder is of great importance:

  • Axial moment of inertia     : I = (π/4)(R⁴ – r⁴)

  • Polar moment of inertia     : J = (π/2)(R⁴ – r⁴)

 Practical   calculation of hollow cylinders

Example 1: Volume calculation

Let’s assume we have a steel pipe with the following properties:

  • Outer radius (R) = 10 cm

  • Inner radius (r) = 8 cm

  • Height (h) = 50 cm

Pipe volume:
V = πh(R² – r²) = π × 50 × (10² – 8²) = 50π × (100 – 64) = 50π × 36 ≈ 5654.87 cm³

Example 2:     Calculating the surface area

For the same steel pipe:

Total area:
A = 2πh(R + r) + 2π(R² – r²)
= 2π × 50 × (10 + 8) + 2π × (100 – 64)
= 100π × 18 + 2π × 36
= 1800π + 72π = 1872π ≈ 5881.06 cm².

Application of a hollow cylinder

1. Pipe production

Examples of hollow cylinders include water, gas, and oil pipes. Their hollow construction reduces weight and saves raw materials.

2. Mechanical engineering

Hollow cylinders are used to manufacture axles, shafts and other rotating parts because this shape has a better moment of inertia than a solid cylinder.

3. Architecture and construction

Hollow columns are used in buildings for aesthetic reasons     and to support loads.

4. Medicine

Hollow cylinders are used in the manufacture of orthopedic implants and surgical instruments.

5. Aerospace industry

The main structures  of rockets  and satellites are typically designed as hollow cylinders to ensure sufficient strength while remaining lightweight.

Advantages of using hollow cylinders

  1. Material savings     : Removing the core saves raw materials.

  2. Lightweight     : lighter than a solid cylinder of similar size.

  3. High strength     : The stress distribution on the wall of the hollow cylinder is uniform.

  4. Thermal efficiency     : In a heating system, the air in the cavities acts as an insulator.

  5. Flexible design     : Allows cables and other pipes to pass through the hollow cylinder.

Disadvantages and limitations

  1. Manufacturing complexity     : Manufacturing a hollow cylinder is more difficult than manufacturing a solid cylinder.

  2. Flexibility:     Can     bend under high pressure.

  3. Restrictions for fish     : Walls that are too thin are not strong enough.

Advanced calculations for hollow cylinders

Pressure in a hollow cylinder  under  internal pressure

Pressurized pipes generate different   pressures   :

  • Loop tension (σθ)     : σθ = (Pᵢ × r² – Pₒ × R² + (Pᵢ – Pₒ)(r²R²)/x²)/(R² – r²)

  • Radial stress (σr)      : σr = (Pᵢ × r² – Pₒ × R² – (Pᵢ – Pₒ)(r²R²)/x²)/(R² – r²)

Where:

  • Pᵢ: internal pressure

  • Pₒ: external pressure

  • x: Distance from the center along the radius

Heat transfer in a hollow cylinder

The heat transfer rate through the wall of a hollow cylinder   is determined by the following formula:

x = (2πkL(T₁ – T₂))/ln(R/r)

Where:

  • k: thermal conductivity coefficient of the material

  • L: Cylinder length

  • T₁, T₂: Temperature of the inner and outer   surfaces

Materials commonly used in hollow cylindrical structures

  1. Metals     : steel, aluminum, copper, titanium.

  2. Polymers     : PVC, HDPE, Polyethylene

  3. Composite materials     : carbon fiber and fiberglass

  4. Ceramics     : High-temperature applications

Design criteria for a hollow cylinder

  1. ASME Boiler and Pressure Vessel Code     : For pressure vessels

  2. ASTM A53     : Standard for Steel Pipe

  3. ISO 2531     : Standard for cast iron pipes

  4. API 5L     : Standard for oil and gas pipelines

Process for the production of hollow cylinders

  1. Rolling mill     : used to produce metal pipes.

  2. Extrusion     : Used to produce polymer profiles.

  3. Centrifugal casting     : for cast iron pipes

  4. 3D printing     : for producing complex patterns

Finally

Hollow cylinders are among the most common geometric shapes in industry and technology     . Due to their numerous advantages, such as material savings, low weight, and high strength, they are widely used in various fields. Understanding the geometric properties, calculation formulas, and applications of these three-dimensional shapes is crucial for engineers and designers. Advances in manufacturing technology are constantly opening up new application possibilities for hollow cylinders, making understanding this geometry even more important.