Washer Volume Calculator
Hollow washer area and volume from outer, inner radius, and height.
What it does
Washers subtract bores from bars: ring area is outer circle minus inner, times height for volume. The same subtraction integrates curved solids of revolution slice by slice.
The tool above takes both radii with height. It returns ring area with volume and void percent.
The default 5/3/10 cm gives 50.27 cm2 and 502.65 cm3 at 36% void. The imperial mirror of 1.97/1.18/3.94 in reads 7.82 in2 and 30.8 in3.
Method follows OpenStax slicing with MIT washer notes and Pappus moments behind them. All three agree on pi(R^2-r^2)h used here.
How to use
- Pick centimeters or inches.
- Enter outer radius R.
- Enter inner radius r.
- Enter height h.
- Read ring area first.
- Read volume beside it.
Formula
- R
- Outer radius.
- r
- Inner radius, 0 for solid.
- h
- Washer height or length.
All inputs share one unit; imperial inches convert to cm first (x 2.54) for shared math. Inner must sit below outer; r = 0 yields solid cylinders. Pappus torus values cross-check the method.
Worked example
Example: 5/3/10 washer
Metric defaults: R 5 cm, r 3 cm, h 10 cm.
- Radii: R 5 cm, r 3 cm, height 10 cm.
- Squares: 25 - 9 = 16.
- Area: 16 x pi = 50.27 cm2.
- Volume: 50.27 x 10 = 502.65 cm3.
- Void: (3/5)^2 = 36%.
- Solid check: r 0 gives pi x 25 x 10 = 785.4.
- Tool shows 50.27 cm2, 502.65 cm3, 36% void.
Reference table
| R / r / h | Area | Volume |
|---|---|---|
| 5 / 3 / 10 | 50.27 | 502.65 |
| 5 / 0 / 10 | 78.54 | 785.40 |
| 10 / 6 / 5 | 201.06 | 1005.31 |
Limits
- Straight walls only.
- Pappus needs clear axes.
- Units must match.
- Negative inputs reject.
Common mistakes
- Using diameters as radii.
- Forgetting to square first.
- Mixing cm with inches.
- Allowing r past R.
- Rounding pi too early.
Try it yourself
Quick checks. Answers sit tucked under each question.
R 5, r 0, h 10?
pi x 25 x 10 = 785.4 cm3 solid.
R 10, r 6, h 5?
A pi x 64 = 201.06, V x 5 = 1005.31 cm3.
R 5, r 4, h 10?
A 28.27, V 282.74 cm3, 64% void at r/R 0.8.
Questions
What is the washer formula?
Ring area pi x (R^2 - r^2) times height. At r = 0 it is a solid cylinder.
Does calculus agree?
Yes: V = integral pi(f^2 - g^2)dx generalizes the straight case to curves.
What is the Pappus check?
Area times centroid travel: a circle radius 2 at distance 4 gives 32 x pi^2.
Does Pappus always apply?
No: the axis must miss the region or Pappus fails.
What if r equals R?
0 volume: inner meets outer, so nothing remains.
How should I round?
Yes: keep pi symbolic to the end, then round area and volume to 2 decimals.