What a fishmouth actually is
When a round tube lands on another round tube, the end of the branch has to be cut to the shape of the surface it is meeting. That shape is the intersection curve of two cylinders, and it is called a fishmouth, a saddle or a cope depending on which trade you learned it in. It is not a simple curve and it cannot be approximated by an arc except when the tubes are very different in size.
The geometry is easier than the name suggests. Put the main tube along an axis and describe its surface as all points at radius R from that axis. Walk around the circumference of the branch tube, and at each position ask how far along the branch you can travel before you hit the main tube surface. That distance changes as you go around, and the difference between the longest and the shortest is the depth of the notch. Write those distances down at regular intervals and you have a template.
For a square tee, the longest reach is on the two sides that lie along the main tube axis, and the shortest is at the top and bottom of the saddle where the branch sits on the crown. For an angled joint the picture rotates: there is one long point, the heel, on the obtuse side, and one short point, the toe, on the acute side. This page indexes stations from the heel because the heel is the point you can find and mark on a real tube without any setting out.
Two ways to transfer the numbers to metal
Both start by finding the heel. On an angled joint that is the side of the tube that will end up furthest from the main tube along the branch axis, which is the obtuse side of the joint. Mark a line down the tube there with a straightedge or a piece of angle iron, then mark the toe on the opposite side.
The wrap method. Cut a strip of paper or thin card long enough to go round the tube, wrap it, and mark the circumference. Divide the half circumference into as many equal steps as you chose stations. Wrap it back on the tube with the seam on the heel line, transfer the divisions to the metal with a marker, then measure each depth back from the end along the tube and connect the dots. This is the method to use when you have no rotary equipment, and the arc distances in the table are what it needs.
The rotation method. If the tube is in a chuck, a rotary table or a marked collar, index by angle instead of by distance. Set the angle mode on this page and the middle column becomes degrees of rotation between readings. The depths are unchanged.
Either way, cut close to the line with a bandsaw or a grinder in a series of straight relief cuts, then blend to the line with a grinder or a half round file. Straight cuts between stations and then blending is faster and more accurate than trying to follow the curve continuously.
What changes with the diameter ratio
| Branch over main | What the notch looks like |
|---|---|
| Under about 0.4 | A shallow saddle. The curve is gentle, few stations are needed and a hole saw at the right angle gets close. |
| 0.4 to 0.9 | The general case, and the one this page is for. The curve changes quickly near the toe and station spacing matters. |
| Above about 0.95 | The intersection approaches a plane. At exactly equal diameters it is a plain miter and a tilting bandsaw vice does it in one cut. |
| Branch larger than main | Not a fishmouth. The larger tube cannot saddle onto the smaller one, and the joint needs a different detail entirely. |
The last row is why this page refuses a branch larger than the main tube. There is no cut on the end of the big tube that makes it sit on the small one, and what people actually build in that situation is a plated or gusseted joint, or they reverse which tube is continuous.
Acute angles and the feather edge problem
As the intersection angle gets smaller, the heel of the notch becomes a long shallow wedge, and the wall of the tube presents itself to the joint at an increasingly oblique angle. Below about 45 degrees the heel is a knife edge with almost no metal behind it. It burns back as soon as an arc reaches it, so there is nothing to fuse to and nothing carrying the joint at exactly the point that is loaded hardest.
Everyone who builds tube structures learns this, and the answers are all detailing answers rather than cutting answers: stop the branch short of the knife edge and fill the resulting gap, gusset the acute corner, or change the geometry so the angle is less severe. What none of them is is grinding the feather edge sharper. If the joint carries load, the detail is a design decision, not a fabrication one, and it belongs with whoever specified the structure.
Checking before you cut
Print or draw the profile on paper, wrap it on the tube, and offer the tube up to the joint before you cut metal. Paper costs nothing and it catches the two errors that actually happen: the angle being measured from the wrong reference, and the heel being marked on the wrong side of the tube. Both produce a notch that is a perfect mirror image of the one you needed.
Round tube out of a rack is also rarely as round as the arithmetic assumes, particularly thin wall tube that has been in a vice. Measure the actual outside diameter in two directions before you trust a template made from a nominal size, and if the two readings differ by more than a wall thickness, expect to fettle the fit.
For the rest of the fabrication, the metal weight calculator handles what the tube weighs, the miter angle calculator handles the flat frame joints that go with the tube joints, and the weld distortion calculator covers the part where a nicely fitted tube frame pulls itself out of square as it cools. The hazards of the welding are named in full on the travel speed calculator.
Questions people ask
Why not just use a hole saw?
For a square tee with a branch much smaller than the main tube, a hole saw in a notching fixture works well and is faster than any layout method. It gets harder as the branch approaches the main tube in diameter, because the saw has to cut a shape that is deeper than its own radius, and it stops working at an angle because a hole saw cuts a circular hole and an angled intersection is not circular. What it produces at an angle is close enough for a light frame and visibly wrong on anything where the fit matters. Layout and grind is slower, works for any combination, and does not need a notcher.
Does wall thickness change the profile?
Not the profile you cut, which is the outside of the branch meeting the outside of the main. It changes what happens at the edge of that cut. On a thin wall tube the cut edge is nearly a line and the fit-up gap is small everywhere. On heavy wall tube, the inside of the branch wall intersects the main tube on a different curve from the outside, so the cut face is a surface rather than an edge and the fit is looser at the toe. That gap is normal, it is what the weld fills, and a consistent small gap is easier to weld than an inconsistent tight one. The wall thickness field on this page is only used to warn about the feather edge at acute angles.
What if the branch does not pass through the centre of the main tube?
That is an offset joint and it is a genuinely different calculation, because the profile is no longer symmetric about the heel and the two halves of the notch differ. This page refuses a non-zero offset rather than quietly returning the symmetric answer, because a template that is silently wrong on one side is worse than no template. Offset joints turn up in real structures, and the practical shop answer without CAD is to make a paper pattern by trial: wrap card, offer it up, trim, repeat. It takes three or four goes and it works.
How many stations do I actually need?
Twelve over the half circumference, which is every fifteen degrees, is a reasonable working minimum for a joint that has to fit. Six is enough to understand the shape and not enough to cut to, because the curve changes fastest near the toe and coarse spacing loses exactly that part. Going to thirty-six is worth it when you are printing a wrap-around template rather than marking stations by hand, since more points cost nothing on paper. On a small branch into a much larger main tube the curve is gentle and fewer stations are fine.
Can I use this for square tube?
No. Every equation on this page assumes both tubes are circular, and a square tube meeting a round one or another square one is a different geometry with a different solution. Square to square at 90 degrees is usually a straight cut, and square to square at an angle is usually a compound miter, which the miter angle calculator handles. Square tube onto round tube needs the corners of the square profile individually developed, which is genuinely awkward by hand and is the case where a paper trial fit beats arithmetic.