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 the annulus area times the length, an annular cross-section extruded along its axis.

Worked example: copper pipe weight Type L 1-inch copper tube has an outside diameter of 1.125 in (R = 0.5625 in) and a 0.050 in wall, which puts the inside diameter at 1.025 in (r = 0.5125 in). For a 10 ft length, h = 120 in:
V = π × (0.5625² − 0.5125²) × 120 = π × 0.05375 × 120 ≈ 20.26 in³ of copper.
Copper runs 8.96 g/cm³, or 0.3237 lb/in³, so 20.26 × 0.3237 ≈ 6.56 lb per 10 ft section.

Tube catalogs list Type L 1-inch at 0.654 lb per foot, which is 6.54 lb per 10 ft. The 0.3% gap is the density figure, not the geometry, and a match that close tells you the wall thickness is right.

Watch that part, because it is where these estimates usually go wrong. Type M in the same nominal size has a 0.035 in wall and weighs 4.65 lb per 10 ft, nearly 30% less. Pull the wrong type off the rack and every weight and material figure moves with it.

Two volumes to distinguish:

1. Wall material volume. What you would weigh on a scale and what you 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 the second one. People asking “how much material is in this pipe?” mean the first. They are very different numbers, so the calculator prints both.

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.

The unrolled view, and why it is not an approximation:

Factor the difference of squares. π(R² − r²)h = π(R + r)(R − r)h = 2π × R_avg × t × h, where R_avg is the mean radius and t = R − r is the wall thickness.

Plenty of references present that second form as a thin-wall shortcut. It is not a shortcut, it is an identity, exactly true at any wall thickness. Which makes it genuinely useful: it gives you a second, independent route to the same number. Picture slitting the pipe lengthwise and flattening it out, and you get a rectangle 2π × R_avg long (the mean circumference), t thick and h tall.

Schedule 40 PVC 4-inch: OD 4.500", ID 4.026", wall 0.237", mean radius 2.1315". For a 10 ft length:
Difference of squares: π × (2.25² − 2.013²) × 120 = π × 1.010331 × 120 ≈ 380.9 in³.
Unrolled rectangle: 2π × 2.1315 × 0.237 × 120 ≈ 380.9 in³.

Identical, because they are the same expression written two ways. If your two answers disagree, one of the four measurements is wrong, and that is worth more than any decimal place.

Sanity check:

• r = 0: V = π × R² × h (a solid cylinder). ✓
• r = R: V = 0 (a zero-wall pipe is just a hole). ✓
• Double the length, double the material. Double the wall thickness on a thin-wall pipe and the material roughly doubles too, since t sits on its own in the unrolled form. ✓