Piping
Elbow cutting formula
In short
To cut a bought elbow to angle α, the cutting plane passes through the bend centre. From the kept face, mark the arcs Le = (R + D/2) × α at the back, Lf = R × α at the side and Li = (R − D/2) × α at the belly, with α in radians.
Calculation
Example calculated with the tool’s default values.
Inputs
- Pipe
- DN50 · 60.3 × 2.9 mm
- Required angle
- 45°
- Elbow radius
- Bought long-radius elbow (EN 10253-2 3D or ASME B16.9 LR per pipe)
- Elbow to cut from
- 90° elbow
- Marks around
- 16
Results
- Length at the back (outside)
- 83.4 mm
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Length at the belly (inside)
- 36.0 mm
- Length at the side (centreline arc)
- 59.7 mm
- Outside chord (rule check)
- 81.2 mm
- Take-out (vertex to face)
- 31.5 mm
- Offcut angle
- 45.0°
- Radius used
- 76.0 mm
- Elbow to cut from
- 90°
- An elbow cannot be flattened: use the template to place the marks; transfer each length on its generator with a flexible rule, then join the points.
Step-by-step calculation
Length at the back (outside)
Le = (R + D/2) × α
106.2 × 0.7854 = 83.4 mm
Length at the side
Lf = R × α
76.0 × 0.7854 = 59.7 mm
Length at the belly (inside)
Li = (R − D/2) × α
45.9 × 0.7854 = 36.0 mm
Mark no. k
φ = 360° (k − 1) / n ; x = (D/2) φ ; l = (R + D/2 cos φ) × α
n = 16
Outside chord
c = 2 (R + D/2) sin(θ/2)
2 × 106.2 × sin 22.5° = 81.2 mm
Take-out
E = R tan(θ/2)
76.0 × tan 22.5° = 31.5 mm
Marks to transfer from the kept face
| No. | Angle from the back (°) | Position around (mm) | Length on the generator (mm) |
|---|---|---|---|
| 1 | 0.0 | 0.0 | 83.4 |
| 2 | 22.5 | 11.8 | 81.6 |
| 3 | 45.0 | 23.7 | 76.4 |
| 4 | 67.5 | 35.5 | 68.8 |
| 5 | 90.0 | 47.4 | 59.7 |
| 6 | 112.5 | 59.2 | 50.6 |
| 7 | 135.0 | 71.0 | 42.9 |
| 8 | 157.5 | 82.9 | 37.8 |
| 9 | 180.0 | 94.7 | 36.0 |
| 10 | 202.5 | 106.6 | 37.8 |
| 11 | 225.0 | 118.4 | 42.9 |
| 12 | 247.5 | 130.2 | 50.6 |
| 13 | 270.0 | 142.1 | 59.7 |
| 14 | 292.5 | 153.9 | 68.8 |
| 15 | 315.0 | 165.8 | 76.4 |
| 16 | 337.5 | 177.6 | 81.6 |
How to use it
- Pick the pipe: outside diameter and wall thickness.
- Enter the required angle (1° to 90°).
- Pick the elbow radius (1.5 D, 1 D, EN 10253-2, ASME B16.9 or custom) and the elbow to cut from: 90°, 45° or the smallest that fits.
- Pick the number of marks around (8, 12, 16, 24 or 32).
- Read the length at the back (outside).
Frequently asked questions
What is the elbow cutting formula?
The cut is radial: its plane passes through the bend centre of the elbow. From the kept face, mark the arc length of each generator: back Le = (R + D/2) × α, side Lf = R × α, belly Li = (R − D/2) × α, with α in radians. For mark no. k, φ = 360° (k − 1) / n, x = (D/2) φ around the pipe and l = (R + D/2 cos φ) × α.
How do I cut a 90° elbow to 45°?
Example: long radius 90° elbow on a 60.3 pipe (R = 90.4 mm), required angle 45°, that is 0.7854 rad. Back: 120.6 × 0.7854 = 94.7 mm; side: 90.4 × 0.7854 = 71.0 mm; belly: 60.3 × 0.7854 = 47.4 mm. The offcut is 45° too. Rule check: outside chord 2 × 120.6 × sin 22.5° = 92.3 mm.
How do I cut an elbow to any degree?
Enter any angle from 1° to 90°: the same arc formulas apply, and the elbow to cut from is a 90° or 45° elbow. An elbow cannot be flattened: the template places the marks; transfer each length on its generator with a flexible rule from the kept face, then join the points.
What is the take-out of a trimmed elbow?
The take-out from vertex to face is E = R tan(θ/2), with θ the required angle. For an elbow of radius 90.4 mm cut to 45°: 90.4 × tan 22.5° = 37.5 mm. Starting from a bought 45° elbow, check its face is truly at 45°: the published B may differ from R tan 22.5°.
Limits
Indicative results. With 1.5 D or 1 D, the radius is taken from the outside diameter: for a bought elbow, pick the standard elbow (EN 10253-2 or ASME B16.9) or enter the catalogue radius. The radius must exceed half the diameter.
Sources
- Longueur d’arc et coupe radiale : Géométrie : arc = rayon × angle (rad), plan de coupe passant par le centre de courbure
- Coupe radiale d’un coude : Géométrie du tore : génératrice φ = arc de rayon R + r cos φ, longueur (R + r cos φ) × α
- EN 10253-2:2021+A1:2026 (raccords à souder bout à bout) : Coudes 2D/3D (rayon R), tés égaux (F), réductions (L) ; tableaux Zickwolff, consultés le 08/11/2026 (link)