Fishmouth Joint Layout for Different Pipe Diameters: 90° Branch

A fishmouth (also written fish mouth, or called a saddle) is the curved cut at the end of a pipe so that it fits against another pipe. When the branch has the same diameter as the main pipe, the layout of the 90° fishmouth joint is simple and symmetrical. In practice, though, the branch is very often smaller than the pipe it will be welded to (an unequal-diameter tee), and then the cut curve of the fishmouth changes.

In this article we work with one specific situation:

• a branch with a diameter equal to or smaller than the main pipe;

• axes that intersect, with no offset;

• a 90° angle between the axes;

• the pattern for the end of the branch, the pipe that will be fitted.

Another angle, offset axes or a branch larger than the main pipe require a different layout. The opening in the main pipe, where the branch will be welded, also has its own layout and is not covered here.

What changes in the fishmouth when the diameters are different

With equal diameters, the intersection is made of two ellipses, each lying in a flat plane, and shows up as straight lines in the front view (elevation). With different diameters that stops being true: the intersection is a three-dimensional (space) curve that no longer lies in a single plane, and the depth has to be calculated on each element line (often just called an element): each of the straight lines that run along the surface of the branch, parallel to its axis.

Let R be the radius of the main pipe and r the radius of the branch (r ≤ R). Measured from the square-cut end of the branch, which touches the sides of the cut, the depth on each element line is:

h = √(R² − r² × sin²θ) − √(R² − r²)

Here, θ is the position of the element line, measured from the plane that contains both axes: 0° and 180° are the element lines lying in that plane (the deepest ones), and 90° and 270° are the sides, where the depth is zero.

Two limiting cases help as a check. If r = R, the formula becomes h = R × |cos θ| and the maximum depth equals the radius, as in the 90° fishmouth article for the same diameter. If the branch is much smaller than the main pipe, the depth tends to zero and the end hardly needs any cut.

Measurements you need

In the example, the main pipe has D = 200 mm (R = 100 mm) and the branch has d = 100 mm (r = 50 mm).

Step-by-step fishmouth layout (parallel-line development)

Step 1 — Draw the views

Draw the front view (elevation): the main pipe horizontal and the branch vertical, with the axes crossing at 90° and the outlines at the actual diameter. Lined up with the branch, draw the top view (the circle of radius r). Next to the front view, draw the end view of the main pipe (the circle of radius R), centered on the extension of the main pipe's axis.

Step 2 — Divide the branch circumference

Divide the circle in the top view into 12 equal parts of 30° and number the points from 0 to 11. Point 0 lies in the plane of the two axes. So points 0 and 6 are the deepest element lines, and points 3 and 9 are the sides.

Step 3 — Transfer each distance to the end view

Each point in the top view marks an element line of the branch: the divisions of the circle are projected by lines parallel to the branch axis, and each one is at a distance z from the plane of the axes (z = r × sin θ). In the top view, the plane of the axes shows up as a straight line (the line of the main pipe's axis) through the center of the circle, and z is the distance from each point to that line. In the end view of the main pipe, draw a vertical line at that same distance z from the center. The point where it meets the circle of radius R, on the upper side, gives the height y of the intersection.

Step 4 — Set the baseline and measure the depths

The vertical through the sides (z = r) meets the circle at a height y₀ = √(R² − r²) = √(10,000 − 2,500) = √7,500 = 86.60 mm. This is the baseline: it coincides with the square-cut end of the branch. The depth of each point is the difference between its height y and the baseline.

Figure 1: Construction drawn to scale. Top: the top view of the branch (radius r) with its 12 points; the plane of the axes is the dashed line. Bottom: the upper half of the end view of the main pipe (radius R). The vertical line from each point, at a distance z from the plane of the axes, meets the circle at height y, and the depth h is the difference from the baseline y₀. Point 1 is highlighted.

Step 5 — Check with the formula

For point 1 (θ = 30°), r² × sin² 30° = 2,500 × 0.25 = 625. Then y = √(10,000 − 625) = 96.82 mm and the depth is 96.82 − 86.60 = 10.22 mm.

For point 2 (θ = 60°), r² × sin² 60° = 2,500 × 0.75 = 1,875. Then y = √(10,000 − 1,875) = 90.14 mm and the depth is 90.14 − 86.60 = 3.54 mm.

The maximum depth, at points 0 and 6, is 100 − 86.60 = 13.40 mm.

Step 6 — Calculate the stretch-out

The branch circumference is C = π × d. With π = 3.1416 and d = 100 mm, C = 314.16 mm. Divided into 12 parts, each division measures 314.16 ÷ 12 = 26.18 mm.

On paper or cardboard, draw a straight line 314.16 mm long (the stretch-out line), mark the 12 divisions and number them 0 to 11 (point 12 is the same as point 0).

Step 7 — Transfer the depths

Erect a perpendicular at each division. Starting from the stretch-out line, which represents the square-cut end of the branch, measure each point's depth inward, toward the free end, using the table below. Points 3 and 9 sit on the line.

Step 8 — Join the points

Join the points with a smooth, continuous curve. Unlike the same-diameter case, there is no sharp tip here: the curve passes smoothly through points 3 and 9. A tip only appears in the limiting case r = R, and the closer r is to R, the tighter the curve gets near the sides.

Figure 2: The template laid out flat, with the stretch-out C = π × d divided into 12 parts. The dashed line is the square-cut end; the yellow curve runs through the 12 points, and the area between the line and the curve is what gets cut away. The height is enlarged 4 times so the curve is readable; the actual depths are in the table.

Step 9 — Cut square, make the template and check it

Cut the branch square to length L, measured along the axis from the square-cut end. If the drawing gives the dimension from where the axes cross, subtract the baseline: L = dimension − 86.60 mm. Length L must be greater than the maximum depth (13.40 mm in the example); otherwise the cut would use up the whole length of the pipe. In terms of the dimension from the crossing, this means a dimension greater than R (100 mm in the example). Then cut out the template. Mark on the pipe the element line of point 0 (a straight line along the pipe); at assembly, it lies in the plane of the axes. Wrap the template around the pipe, with its base on the square-cut end and point 0 on that element line. Mark the outline and check that points 3 and 9 sit on the sides. Only then cut along the profile. After cutting, check the fit on the main pipe and adjust with a file or grinder if needed.

Measurement table: 100 mm branch on a 200 mm pipe

Values for D = 200 mm, d = 100 mm, axes at 90° and 12 divisions. Depth is measured from the square-cut end of the branch.

The maximum depth is 13.40 mm. For comparison: with a branch of the same diameter as the main pipe (200 mm), it would be 100 mm. If your layout gives different values, check it before cutting.

How the diameter ratio changes the depth

For a 200 mm main pipe, the maximum depth of the branch grows quickly as it approaches the diameter of the main pipe:

Each value comes from R − √(R² − r²). When the branch has the same diameter as the main pipe, the maximum depth equals the radius.

Figure 3: Depth h of the cut along the circumference, with the actual values, for branches of diameter d of 50, 100, 150 and 200 mm on a 200 mm main pipe. The closer the branch diameter gets to the main pipe diameter, the deeper the curve; with d = 200 mm (equal diameters) the sharp tip appears.

What if the part is made from plate?

When the pipes are rolled from plate, use the mean diameter of each one, that is, the outside diameter minus the wall thickness. Example with a 200 mm main pipe with 6 mm wall thickness, and a 100 mm branch with 5 mm wall thickness:

• main pipe: 200 − 6 = 194 mm, or R = 97 mm;

• branch: 100 − 5 = 95 mm, or r = 47.5 mm;

• C = 3.1416 × 95 = 298.45 mm, or 24.87 mm per division;

• maximum depth: 12.43 mm.

This is a common approximation. The weld bevel and the final gap should be adjusted at fit-up.

Common mistakes when laying out a fishmouth between different diameters

1. Using the same-diameter layout

That layout only holds when the diameters are equal. With a 100 mm branch on a 200 mm pipe, using it gives, for the branch, a depth of 50 mm (its radius), and the correct depth is 13.40 mm.

2. Swapping R and r in the formula

The R in the formula is the radius of the main pipe and r is the radius of the branch. Swapping them makes the value under the square root (r² − R²) negative, and a negative number has no real square root.

3. Using a branch larger than the main pipe

The formula only holds for r ≤ R. If the branch is larger, the layout is different.

4. Using the nominal diameter

The layout depends on the actual outside diameter of both pipes. Measure the pipes; do not rely on the nominal diameter alone.

5. Forgetting the opening in the main pipe

The cut end of the branch is only half of the job. The main pipe also needs an opening with its own layout, which is not covered in this article.

6. Measuring depth from another reference

All depths come from the same baseline, the square-cut end. Mixing references shifts the whole cut.

Frequently asked questions

Can I use the 45° rule for different diameters?

No. The u × constant rule (u is the distance from an element line to the branch axis) in the 45 degree lateral layout article holds for the same diameter. For different diameters, use the formula in this article or the general formula with the angle, which is in the frequently asked questions of that article.

How deep is the fishmouth cut on a smaller pipe?

Use h = √(R² − r² × sin²θ) − √(R² − r²), where θ is the position of each element line and the depth is measured from the square-cut end. In the example (200 mm main pipe and 100 mm branch), the maximum depth is 13.40 mm.

What if the axes do not intersect?

If the branch is offset from the axis of the main pipe, the layout is different and is not covered here.

Can I use a different number of divisions?

Yes. The number must be even, so that points 0 and n ÷ 2 (0 and 6 for 12 divisions) exist, both in the plane of the axes, and preferably a multiple of 4 (12, 16, 24), so that the sides, at points n ÷ 4 and 3n ÷ 4, fall exactly on a division. More divisions give you more points to join.

See also

• How to lay out a 90° fishmouth joint in metal fabrication

• 45 Degree Lateral Layout: Step-by-Step Fishmouth Template

Step-by-step summary

To lay out a 90° fishmouth with a branch smaller than the main pipe:

1. draw the front view, the top view of the branch and the end view of the main pipe;

2. divide the branch circumference into equal parts and number the points;

3. transfer each distance z to the end view and find the height y on the circle of radius R;

4. measure each depth from the baseline y₀ = √(R² − r²), or use the formula;

5. calculate the stretch-out C = π × d and divide it into the same number of parts;

6. transfer the depths from the square-cut end;

7. join the points with a smooth curve, with no sharp tip;

8. cut square, make the template, check the marked outline, cut, and only then check the fit.

Important

This article covers specifically the case of a branch with a diameter equal to or smaller than the main pipe, with axes that intersect at 90°. Changing any of these conditions changes the pattern. When cutting the pipe, wear appropriate personal protective equipment and secure the workpiece firmly.