Pipe offset formulas
The geometry behind a simple offset, a rolling offset, the roll angle and the cut length — worked through with real numbers, so you can check any of it against the fitting in your hand.
Written by the developer of Spoolset. Every figure here is the same figure the app produces.
Simple offset — one plane
A pipe run has to step around something. Two fittings of the same angle turn the pipe out and back, and the piece between them is what you cut. Three measurements describe it:
- Offset — how far the run shifts sideways. This is what you measure.
- Travel — the centre-to-centre length of the diagonal piece.
- Run — how far along the original direction the offset consumes.
The diagonal, the offset and the run form a right-angled triangle, so:
| Quantity | Formula |
|---|---|
| Travel | offset / sin(angle) |
| Run | offset / tan(angle) |
In the field nobody reaches for a sine table — you multiply. These are those two divisions worked out for the six fitting angles that actually exist:
| Fitting angle | Travel = offset × | Run = offset × |
|---|---|---|
| 11¼° | 5.1258 | 5.0273 |
| 22½° | 2.6131 | 2.4142 |
| 30° | 2.0000 | 1.7321 |
| 45° | 1.4142 | 1.0000 |
| 60° | 1.1547 | 0.5774 |
| 90° | 1.0000 | 0 |
Two of those are worth committing to memory. At 45° the travel is the offset times 1.4142 and the run equals the offset — the triangle is isosceles, which is why 45s are the fitter's default. At 30° the travel is exactly double the offset, because sin 30° is exactly ½.
Worked example
A 10″ offset using 45° elbows:
- Travel = 10 × 1.4142 = 14.14″, which is 14-1/8″ to the nearest sixteenth.
- Run = 10 × 1.0000 = 10″.
That 14-1/8″ is centre to centre. It is not the length of pipe you cut. See take-outs.
Rolling offset — two planes at once
A rolling offset shifts the run vertically and horizontally at the same time. It is the one that gets people, because the fitting angle no longer sits in the plane you measured in.
Two measurements go in:
- Set — the vertical shift.
- Roll — the horizontal shift.
The trick is that those two combine into a single diagonal — the true offset — and once you have it, a rolling offset is just a simple offset in a tilted plane:
| Step | Formula |
|---|---|
| 1. True offset | √(set² + roll²) |
| 2. Travel | true offset / sin(angle) |
| 3. Run | true offset / tan(angle) |
| 4. Roll angle | atan(roll / set) |
Step 4 is the one most explanations leave out, and it is the number the job actually needs: how far to rotate the fitting off vertical. Get the travel right and the roll angle wrong and the pipe is the correct length pointing the wrong way.
Why you also see √(A² + B² + C²)
On the forums the rolling offset is often given as
travel = √(set² + roll² + run²), described as working for any angle. That is
correct, and it is the same geometry: it is the straight-line distance between two points in
three dimensions. It just needs the run as an input, which you usually do not
know yet — whereas the angle you do know, because it is stamped on the fitting.
The two agree. At 45° the run equals the true offset, so
√(trueOffset² + trueOffset²) = trueOffset × 1.4142 — which is exactly
what step 2 gives. Use the angle when you know the fitting; use the three-dimensional form when
you know the run and want to check yourself.
Worked example
A set of 6″, a roll of 8″, using 45° elbows:
- True offset = √(6² + 8²) = √100 = 10″.
- Travel = 10 × 1.4142 = 14-1/8″ centre to centre.
- Run = 10 × 1.0000 = 10″.
- Roll angle = atan(8 / 6) = 53.1° off vertical.
The 6-8-10 triangle is worth knowing for exactly this reason — it comes out whole.
Take-outs — why the cut length is shorter
Every formula above gives you centre to centre: the distance between the centre points of the two fittings. But a fitting occupies some of that distance. The take-out is the length from the centre of the fitting to its face — the part the fitting itself fills.
So:
| Quantity | Formula |
|---|---|
| Cut length (end to end) | travel − take-out A − take-out B |
This is where pipe gets wasted. Cut to the travel and the assembly is long by the sum of two take-outs — on 2″ 45s that is 2¾″.
ASME B16.9 long-radius elbows, centre to face
These are published dimensions of factory-made butt-welding fittings, not something anyone derives. Cross-checked against two independent references:
| NPS | 45° elbow | 90° elbow |
|---|---|---|
| ½″ | 5/8″ | 1-1/2″ |
| ¾″ | 3/4″ | 1-1/2″ |
| 1″ | 7/8″ | 1-1/2″ |
| 1¼″ | 1″ | 1-7/8″ |
| 1½″ | 1-1/8″ | 2-1/4″ |
| 2″ | 1-3/8″ | 3″ |
| 2½″ | 1-3/4″ | 3-3/4″ |
| 3″ | 2″ | 4-1/2″ |
| 4″ | 2-1/2″ | 6″ |
| 5″ | 3-1/8″ | 7-1/2″ |
| 6″ | 3-3/4″ | 9″ |
| 8″ | 5″ | 12″ |
The 90° column is 1.5 × NPS above 1″, with a 1-1/2″ floor on the
small sizes. The 45° column is not a clean multiple of anything, which is why it has to be
looked up.
B16.9 tabulates 45° and 90° only. There is no standard long-radius elbow at 11¼°, 22½°, 30° or 60°, so there is no published take-out to look up. Those angles come from mitred or cut fittings, and the honest answer is to measure the fitting in your hand.
A take-out can be derived as R × tan(angle / 2) for a bend of radius R, and it
is a reasonable approximation — but it is an approximation, and treating it as a published
figure is how wrong cuts happen. On a ½″ 45° elbow that formula is wrong by a factor of
two against the real B16.9 value.
Worked example, end to end
The 10″ offset from earlier, in 2″ pipe with 45° long-radius elbows:
- Travel = 14-1/8″ centre to centre.
- Take-outs = 1-3/8″ + 1-3/8″ = 2-3/4″.
- Cut length = 14-1/8 − 2-3/4 = 11-3/8″ end to end.
Grade and fall
Drainage is specified as a slope, and it is quoted two different ways depending on who is asking. Both come from the same ratio:
| Quantity | Formula |
|---|---|
| Grade, percent | (rise / run) × 100 |
| Fall, inches per foot | (rise / run) × 12 |
The one to know: ¼″ per foot is 2.08%. It is the standard fall for small-diameter sanitary drainage in much of the trade, and it is a number worth recognising instantly when a drawing quotes a percentage instead.
A 2″ drop over an 8′ run is 2 / 96 = 0.0208 — 2.08%, or ¼″ per foot. The same slope, said twice.
Checking your own numbers
Everything above is arithmetic you can do on any calculator, and it is worth doing by hand at least once — an app that gives you a number you cannot sanity-check is a worse tool than a formula you understand.
Spoolset does these four calculations on an iPhone, draws the shape to scale as you type so a mistyped number shows up as a wrong picture, and cites the source of every table it uses. Simple offsets and grade are free; rolling offsets and the take-out tables are a one-time purchase.
