Hollow Cylinder Volume Calculator (Pipe Wall)

A hollow cylinder is a pipe — a cylinder with a smaller cylinder removed from the inside. The wall material volume (the actual solid part of the pipe) is the outer cylinder minus the inner one:

V = π × (R² − r²) × h

Where R is the outer radius, r is the inner radius, and h is the length (height) of the pipe.

This is also the annulus area × length — annular cross-section extruded along its axis.

Worked example — copper pipe weight: Type L 1-inch copper pipe has OD 1.125 in (R = 0.5625 in) and wall thickness 0.040 in (r = 0.5225 in). For a 10 ft length: V = π × (0.3164 − 0.273) × 120 ≈ π × 0.0434 × 120 ≈ 16.36 in³ of copper. Copper density is 0.323 lb/in³. Weight = 16.36 × 0.323 ≈ 5.28 lb per 10 ft section. Most catalogs list Type L 1-inch at 5.13 lb/10ft — within 3%, the rest being slight wall thickness variance.

Two volumes to distinguish:

1. Wall material volume — what you’d weigh on a scale, what you’d pay for. Use V = π × (R² − r²) × h.
2. Internal (capacity) volume — how much fluid the pipe holds. Use V = π × r² × h.

People asking “what’s the volume of this pipe?” almost always mean #2. People asking “how much material is in this pipe?” mean #1. They’re very different.

Where hollow cylinder volume matters in practice:

• Pipe and tube material cost. Order copper, steel, or PVC by weight. Weight comes from wall volume × density.
• Pipe insulation. A sleeve of insulation around a pipe is also a hollow cylinder. Volume = π × (R_outer_insulation² − R_pipe²) × length.
• Sleeve bearings and bushings. Brass sleeves between rotating shaft and housing.
• Concrete column rebar layout. Cylindrical concrete columns reinforced around a hollow steel core.
• Boilers and pressure vessel walls. Thick-wall vessels designed for high pressure.

For thin-wall pipes (R − r is small compared to R), there’s a useful approximation: V ≈ 2π × R_avg × t × h, where R_avg is the mean radius and t is wall thickness.

This is the “unrolled” view — imagine slitting the pipe lengthwise and flattening it out. You get a rectangle: 2π × R_avg long (the mean circumference), t thick (wall thickness), h tall (pipe length).

For schedule 40 PVC 4-inch pipe: OD 4.500", ID 4.026", wall 0.237". Mean radius 2.13", wall thickness 0.237". For 10 ft length: Exact: π × (5.063 − 4.052) × 120 ≈ 380.9 in³. Approximation: 2π × 2.13 × 0.237 × 120 ≈ 380.5 in³. Within 0.1%.

Sanity check:

• r = 0: V = π × R² × h (a solid cylinder). ✓
• r = R: V = 0 (zero-wall pipe is just a hole). ✓