Fish Mouth Cut Pipe Formula — Layout, Template and Worked Example

The fish mouth cut pipe formula gives the depth of material to remove at each point around a branch pipe's circumference so the cut end sits flush against the curved surface of a larger main pipe. It is the mathematical foundation behind every saddle cut template — whether marked by hand or generated by a notching calculator. This article derives the formula, explains every variable, and works through a real example using IS:1239 pipe OD values.

What Is a Fish Mouth Cut and When Is It Used?

A fish mouth cut — also called a saddle cut or cope cut — is the curved profile cut at the end of a branch pipe where it meets a larger main pipe at an angle. The cut allows the branch pipe end to nest snugly against the curved outer surface of the main pipe, producing a tight joint that requires minimal filler weld and eliminates gaps that cause weld defects. It is standard in handrail fabrication, pipe rack construction, structural tube framing, and process piping across India.

Fish mouth cut ka formula aur template banane ki zaroorat tab padti hai jab branch pipe ka OD main pipe ke OD se chhota ho. The same mathematics applies at any intersection angle — 90°, 45°, or any value — through the sin α term in the full formula.

The Fish Mouth Cut Formula — Variables and Derivation

Full formula (any intersection angle):
z(θ) = r_m / sin α − √(r_m² − (r_b × cos θ × sin α)²) / sin α

Simplified formula (90° T-joint only, sin α = 1):
z(θ) = r_m − √(r_m² − (r_b × cos θ)²)

z(θ)

Cut depth at angular position θ, in millimetres — material to remove from the branch pipe end at that point. Maximum at θ = 0° (crown) and θ = 180° (saddle). Zero at θ = 90° and 270° (the neutral axis sides).

r_m

Radius of the main pipe = OD_main ÷ 2, in millimetres. Always use IS:1239 OD values, not NB sizes.

r_b

Radius of the branch pipe = OD_branch ÷ 2, in millimetres.

α

Intersection angle between branch and main pipe axes, in degrees. For a right-angle T-joint α = 90°, sin α = 1.

θ

Angular position around the branch pipe circumference, 0° at the crown, sweeping to 360°. Calculate at 0°, 30°, 60°, 90°, 120°, 150°, 180° — the second half (180°–360°) mirrors the first.

Cut-Mark Depth Table — Common IS:1239 Branch Sizes onto 100 NB Main at 90°

The table below pre-calculates saddle cut depths at four key angular positions for common branch NB sizes joining a 100 NB main pipe (OD = 114.3 mm) at 90°. Use these values to mark a paper wrap-around template directly on the pipe.

Main pipe: 100 NB, OD = 114.3 mm. Intersection angle: 90°. OD values per IS:1239 (Part 1):2004. Calculated using z(θ) = r_m − √(r_m² − (r_b × cos θ)²). For angles other than 90° or for a full 360° printable template, use our free Pipe Notching Calculator.

Worked Example — 40 NB Branch onto 100 NB Main at 90°

A handrail frame requires a 40 NB Medium Class MS branch pipe to meet a 100 NB Medium Class MS main pipe at 90°. The fabricator needs to mark the saddle cut on the branch pipe end before cutting with an angle grinder.

Inputs from IS:1239 (Part 1):2004:
Main: 100 NB → OD = 114.3 mm → r_m = 57.15 mm
Branch: 40 NB → OD = 48.3 mm → r_b = 24.15 mm
α = 90° → use simplified formula: z(θ) = r_m − √(r_m² − (r_b × cos θ)²)

Calculation at 6 positions:
θ = 0°: z = 57.15 − √(3266.1 − 583.2) = 57.15 − 51.80 = 5.35 mm
θ = 30°: z = 57.15 − √(3266.1 − 437.2) = 57.15 − 53.18 = 3.97 mm
θ = 60°: z = 57.15 − √(3266.1 − 145.8) = 57.15 − 55.86 = 1.29 mm
θ = 90°: z = 57.15 − √(3266.1 − 0) = 57.15 − 57.15 = 0.00 mm
θ = 120°: mirror of 60° = 1.29 mm
θ = 150°: mirror of 30° = 3.97 mm
θ = 180°: mirror of 0° = 5.35 mm

Marking the template: Wrap a paper strip around the 40 NB branch pipe end. The full circumference = π × 48.3 = 151.7 mm. Divide into 6 equal arcs of 25.3 mm. At each division, measure the z value from the pipe end and mark the point. Connect all points with a smooth curve. Cut along this line with a grinder or plasma cutter. The maximum cut depth is 5.35 mm at the crown and saddle points.

For a full SVG template generated automatically for any pipe size and angle from 15° to 90°, use our Pipe Notching / Saddle Cut Calculator. For the broader theory of saddle joints in fabrication, see our Pipe Notching and Saddle Cuts guide.