Annulus Volume Calculator: Formula and Examples

An annulus volume calculator finds the volume of a hollow cylinder, such as a pipe, sleeve, or cylindrical tank wall. Enter the outer radius, inner radius, and length using the same units; the result is the material volume between the two circular surfaces.

Formula

The annulus volume is the volume of the full outer cylinder minus the volume of the inner cylinder:

V = πL(R² − r²)

• V is annulus volume, in cubic units such as cm³, m³, in³, or ft³.
• R is the outer radius, measured from the cylinder’s central axis to its outside surface.
• r is the inner radius, measured from the same axis to the inside surface.
• L is the cylinder’s length or height, measured parallel to its axis.
• π is approximately 3.14159265.

The calculator’s inputs and outputs use centimetres and cubic centimetres. Use the formula with any consistent length unit: for example, metres produce cubic metres, while inches produce cubic inches. The geometric requirements are L ≥ 0 and R ≥ r ≥ 0. A real hollow cylinder has R greater than r.

Worked Example

Using the calculator’s default values, the outer radius is 5 cm, the inner radius is 3 cm, and the length is 10 cm.

1. Square each radius: R² = 5² = 25 cm² and r² = 3² = 9 cm².
2. Subtract the inner cross-sectional area factor: R² − r² = 25 − 9 = 16 cm².
3. Multiply by the length and π: V = π × 10 × 16 = 160π cm³.
4. Evaluate and round to two decimal places: V ≈ 502.65 cm³.

As a check, the outer cylinder is 250π ≈ 785.40 cm³ and the hollow inner cylinder is 90π ≈ 282.74 cm³. Their difference is 160π ≈ 502.65 cm³.

When to Use an Annulus Volume Calculator

Use this calculation when the object has a constant circular cross-section and a straight central opening extending through its full length. It gives the volume of the ring-shaped material, not the capacity of the hollow space. Typical geometric examples include a uniform pipe wall, a cylindrical spacer, or a straight hollow rod.

It can also help estimate material quantity from ideal dimensions, but the answer is only a geometric volume. It does not by itself give mass, because mass depends on the material’s density, nor does it account for manufacturing tolerances, fittings, end caps, or irregular surfaces.

How to Measure the Inputs

Measure diameters or radii carefully and distinguish between the inside and outside dimensions. If you measure diameters, convert them to radii by dividing by two: R = outside diameter ÷ 2 and r = inside diameter ÷ 2. Alternatively, use the equivalent diameter formula V = (πL/4)(D² − d²), where D and d are the outer and inner diameters.

• Measure the length along the cylinder’s axis, not around its circumference.
• Use either all radii or all diameters in the corresponding formula; do not enter a diameter as though it were a radius.
• Keep all measurements in one unit before calculating. Convert inches to feet or millimetres to metres first if needed.
• For an actual object, take measurements at more than one point if the wall may be uneven or the cylinder may be tapered.

Reading and Converting the Result

A cubic unit describes three-dimensional space. The output of 502.65 cm³ is equivalent to about 0.50265 litres, since 1,000 cm³ = 1 litre. In US customary units, 1 cubic foot equals 1,728 cubic inches; volume conversions must cube the length conversion factor. For instance, converting a result in cubic inches to cubic feet means dividing by 1,728, not by 12.

The calculator also displays the outer cylinder volume and inner hollow-space volume as useful checks. The annulus result should equal the outer volume minus the inner volume. If it does not, check the entered measurements and units. These helper values describe ideal cylinders with the same length as the annulus.

Common Errors and Limits

A frequent mistake is subtracting radii before squaring them. The cross-sectional area is πR² − πr², not π(R − r)². For example, the default radii give a squared-radius difference of 25 − 9 = 16, whereas squaring the wall thickness would give (5 − 3)² = 4 and substantially understate the volume.

The formula assumes straight, parallel sides and a circular opening centered along the same axis as the outside surface. It does not directly describe a bent pipe, a tapered sleeve, an off-center bore, or a cylinder with changing dimensions. For a variable shape, divide it into suitable sections and calculate each section separately, or use a method designed for that geometry.

Rounding can also affect results. Keep full precision during intermediate steps and round only the final displayed value. Small measurement errors in the radii can have a noticeable effect, particularly when the wall is thin, because the calculation subtracts two similar cylinder volumes.

Frequently Asked Questions

What is the formula for the volume of an annulus?
For a hollow cylinder with outer radius R, inner radius r, and length L, use V = πL(R² − r²). Subtract the inner cylinder’s volume from the outer cylinder’s volume.

Can I use diameter instead of radius?
Yes. Use V = (πL/4)(D² − d²), where D is the outside diameter and d is the inside diameter. Do not put diameters into the radius formula without dividing each by two.

Is annulus volume the same as pipe capacity?
No. Annulus volume is the material in the pipe wall. The pipe’s internal capacity is πr²L, using the inner radius only.

How do I find the mass of a hollow cylinder?
Multiply its material volume by the material density, using compatible units. This calculator supplies the geometric volume; density must be known separately, and real objects may differ from the ideal shape.

What if the inside and outside measurements are reversed?
The inner radius cannot exceed the outer radius. If r is greater than R, the formula returns a negative difference, indicating that the dimensions were entered in the wrong fields or are inconsistent.

Related

• Calculate the Hypotenuse of a Triangle
• Absolute Minimum and Maximum Calculator Explained
• Calculate the Golden Ratio

in outer_radius | Outer radius | cm | 5 | 0
in inner_radius | Inner radius | cm | 3 | 0
in length | Length | cm | 10 | 0
out volume | Annulus volume | cm³ | pi * length * (outer_radius^2 - inner_radius^2) | 2
out outer_volume | Outer cylinder volume | cm³ | pi * length * outer_radius^2 | 2
out inner_volume | Hollow space volume | cm³ | pi * length * inner_radius^2 | 2