Hollow cylinder (pipe)

Properties

• It has an outer curved surface, an inner curved surface and two ring-shaped ends.
• The volume of the material is the outer cylinder minus the inner one.
• The space inside, the capacity, depends only on the inner radius.

Why the formulas work

The end of the tube is a ring of area π(R² − r²), and the tube is that ring carried along the length h, so the volume of material is π(R² − r²)h. The curved surface has two parts, outside 2πRh and inside 2πrh. The two ends are rings.

Worked example

Take a hollow cylinder (pipe) with outer radius 7 cm, inner radius 5 cm and height or length 20 cm.

1. Volume of material. Volume = π(R² − r²)h. Here: π × (7² − 5²) × 20 = 1,507.96 cm³ (with π = 22/7: 1,508.57 cm³).
2. Curved surface area (outer + inner). Curved surface area = 2π(R + r)h. Here: 2 × π × (7 + 5) × 20 = 1,507.96 cm² (with π = 22/7: 1,508.57 cm²).
3. Total surface area. Total surface area = 2π(R + r)h + 2π(R² − r²). Here: 1,507.96 + 2 × π × (7² − 5²) = 1,658.76 cm² (with π = 22/7: 1,659.43 cm²).
4. Capacity (space inside). Capacity = πr²h. Here: π × 5 × 5 × 20 = 1,570.8 cm³ (with π = 22/7: 1,571.43 cm³).

Finding a missing measurement

The thickness of the wall is R − r. If the outer radius is 7 cm and the wall is 2 cm thick, the inner radius is 5 cm. To find the weight of a pipe, multiply the volume of the material by the density of the metal. The capacity is the figure to use only for what flows inside.

Mistakes to avoid

• Mixing up the volume of the material (for weight or cost of metal) with the capacity (for how much water it holds).
• Using the wall thickness as the inner radius. The inner radius is the outer radius minus the thickness.

Where you see it

Water pipes, drinking straws, a roll of tape, bamboo, rubber hoses and well rings are hollow cylinders.

Try these

Work each one out, then tap to check.

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