Subtract the empty bore from the outer solid
For a concentric cylindrical shell with outside radius R, inside radius r and shared axial length h, material volume is V=πh(R²−r²). This is the outer full cylinder minus the inner empty cylinder. Require 0<r<R for a genuinely hollow shell. The formula assumes the bore extends through the full shared length and describes ideal geometry. It does not automatically include a bottom, cap, threads, flanges or other features that alter the material shape.
Identify radii rather than diameters
If outside diameter is D and inside diameter is d, use R=D/2 and r=d/2. The equivalent diameter formula is V=πh(D²−d²)/4. Both diameters must refer to the same cross-section and use the same measurement system. Nominal pipe labels do not necessarily equal measured outside or inside diameters, so obtain actual dimensional specifications. Substituting a nominal label as an exact geometric measurement can produce an incorrect result even when the subtraction formula is implemented perfectly.
Work an original shell example
Let outside diameter be 10 cm, inside diameter 8 cm and length 20 cm. Radii are 5 and 4 cm. Material volume is π×20×(25−16)=180π cm³, approximately 565.4866776 cm³. The outer cylinder volume is 500π cm³ and the bore volume is 320π cm³. Their difference is the same 180π. This two-run check keeps empty space and solid material visible rather than labeling one unexplained output as cylinder volume.
Relate wall thickness to the inside diameter
For uniform radial wall thickness t, the inner radius is R−t and inner diameter is D−2t. In the example, t=1 cm reduces outside diameter 10 to inside diameter 8, not 9. Thickness is present on both sides of the cross-section. Verify that t<R before using the hollow model. If thickness reaches the radius, the bore disappears; beyond that, a negative inner radius is not a physical correction and should not be squared into a plausible-looking result.
Keep bore capacity separate from material volume
The example's bore encloses 320π cm³, approximately 1,005.309649 cm³ or 1.005309649 L. That is different from the shell's 565.4866776 cm³ of material. A larger bore can increase capacity while reducing material for the same outer dimensions. State which quantity is being estimated. Neither geometric volume determines mass without a relevant density, and density units must match the chosen volume unit. The general Volume Calculator does not supply a verified material-density or manufacturing-tolerance model.
Handle closed ends with separate geometry
A through-bore shell has no bottom in this model. If an actual cup has outer height 20 and bottom thickness 1, its inner cavity might have height 19 under a specifically defined flat-bottom construction. Then subtract inner πr²×19 from outer πR²×20, giving 196π for R=5 and r=4. This is not the same as adding a full outer-radius disk to the through-shell without checking overlap. Define the outer envelope and the actual cavity before subtracting.
| Quantity | Exact expression | Meaning |
|---|---|---|
| Outer full cylinder | 500π cm³ | Outside envelope |
| Through-bore | 320π cm³ | Empty interior |
| Through-shell material | 180π cm³ | Outer minus bore |
| Defined 1-cm-bottom cup | 196π cm³ | Outer minus 19-cm-high cavity |
Outside radius 5, inside radius 4; outer height 20. The final row changes the cavity height explicitly.
Use separate supported cylinder runs
Select Cylinder in the Volume Calculator. Enter radius 5 and height 20 for the outside, then radius 4 and height 20 for the through-bore. Subtract the results at retained precision. The tool has no dedicated hollow-cylinder mode, so the manual subtraction and shared-height assumption remain part of the workflow. If starting from diameters, halve them before each radius entry. Changing only a unit label does not convert an existing numerical input to another length unit.
Check a thin-wall approximation cautiously
For a thin shell, an approximate volume can be written as circumference at a chosen reference radius times thickness times length. Choosing the mean radius (R+r)/2 actually reproduces π(R+r)(R−r)h exactly for this ideal uniform concentric shell. Using outside circumference 2πR instead overestimates by πht². Identify the reference radius rather than quoting an unlabeled circumference-times-thickness shortcut. Irregular thickness or eccentric bores require additional measurements and may not satisfy the simple annular model.
Save the assumptions with the result
Record outside and inside dimensions, whether they are radii or diameters, axial lengths, end conditions and measurement units. Converting centimeters to meters requires a cubic factor of one million for volume. Keep tolerances separate from final decimal formatting; subtracting nearly equal large volumes can make a thin-shell estimate sensitive to measurement error. The difference-of-squares relationship can also be evaluated as (R−r)(R+r). That algebra helps numerical calculation but cannot recover missing precision in the original measurements.