How to lay out a square to round transition

Guide · Published 1 October 2026 · Worked examples are calculated with the square to round transition calculator

A square to round (or rectangle to round) transition is laid out by triangulation: four flat triangles, one on each side, and four cone-like corner fans that are split into thin triangles. Find the true length of every line, lay the triangles down edge to edge, and the pattern appears. Here is the method, with a fully worked 500 mm square to 300 mm round.

What the surface is made of

Four flat triangles. Each side of the square is the base of a triangle whose apex touches the round where the round is tangent to that side. These are flat, so they bend only along their edges.
Four corner fans. Between two side triangles, the surface runs from one corner of the square to a quarter of the round. It is part of an oblique cone with its apex at the corner, so it rolls smoothly. Dividing that quarter into equal parts and joining each point to the corner splits the fan into narrow triangles.

That is why a square to round has straight bend lines fanning out from each corner, and why its pattern is built from triangles rather than arcs.

Plan and front view of the example: 500 × 500 mm base, 300 mm round, 350 mm high, each quarter of the round in 3 divisions, seam up the middle of the front side.

True lengths: the only formula you need

Every line on the surface rises by the full height H, so its true length is √(H² + plan length²), where the plan length is what you measure in the view from above. Two lines in the example can be checked by hand:

Height of a side triangle, square to the base side: in plan it runs from the middle of the side (250 mm from the centre) to the round (150 mm from the centre), 100 mm. True length √(350² + 100²) = 364.0 mm.
Corner to the tangent point (the edge shared by a fan and a side triangle): in plan √(250² + 100²) = 269.3 mm, so true length √(350² + 269.3²) = 441.6 mm.

The calculator finds the rest. Point 0 is the tangent point of the side before the corner; point 3 is the tangent point of the side after it.

True lengths from each corner to the division points on the round (mm)
Point	From corner A	From corner B	From corner C	From corner D
0	441.6	441.6	441.6	441.6
1	409.3	409.3	409.3	409.3
2	409.3	409.3	409.3	409.3
3	441.6	441.6	441.6	441.6

Sides (mm)
Side	Base length	Slant height (square to the side)	Seam line
front	500.0	364.0	364.0
right	500.0	364.0
back	500.0	364.0
left	500.0	364.0

With the round centred on a square base, all four corners have the same fan and every fan is symmetric: the lines to the two tangent points are the longest (441.6 mm) and the lines to the middle points the shortest. The chord between neighbouring division points on the round is 77.6 mm.

Laying the pattern out

Start at the seam. With a seam up the middle of the front side, draw the seam line: 364.0 mm, from the middle of the base side up to the round.
From the bottom of the seam line, mark half the base side (250 mm) square to it along the base edge: that is corner A (front-left). Check that the line from corner A to the top of the seam equals the corner-to-tangent-point length in the table.
Fan round corner A: from the corner swing each true length in turn, and from the previous point on the round swing the division chord (77.6 mm). Each crossing is the next point on the round.
After the last fan line, lay the next side triangle: its base is the full side (500 mm) and its two other edges are the corner-to-tangent-point lengths of the two corners at its ends.
Carry on round the four corners and back to the seam, then draw the round edge as a smooth curve through the points.
Add seam, base and top allowances, and mark the bend lines (the corner and fan lines) on the outside.

cut line
bend lines (corner and fan lines)

The finished pattern of the example from the square to round calculator. The dashed bend lines are the triangulation lines.

Enter your own base, round and height to get the same tables and a full-size template on A4 or Letter paper, or DXF and SVG files for cutting.

Open this square to round in the calculator

How many divisions?

More divisions per quarter give a smoother round edge and more bend lines. For hand layout, 3 or 4 per quarter (12 or 16 round the circle) is common for small duct work; larger transitions and plate work need 6 or more so the straight chords do not show. The development is exact at the division points, so the choice changes how many lines you mark, not the accuracy of the points.

Rectangle to round

Nothing changes in the method, except that opposite sides have different triangles and the fans are no longer symmetric: the long side's triangle is wider and shorter. Enter the rectangle's width and depth and the table gives a column per corner as before.

Offset square to round

When the round is not centred over the base (for an offset or a flat side), each corner gets its own fan and the side triangles lean. The method is identical: every plan length changes, so every true length changes. With the same example moved 75 mm to the right:

True lengths with the round offset 75 mm to the right (mm)
Point	From corner A	From corner B	From corner C	From corner D
0	464.4	403.9	430.8	488.0
1	437.2	383.3	393.9	446.6
2	446.6	393.9	383.3	437.2
3	488.0	430.8	403.9	464.4

Corners B and C (front-right and back-right, on the side the round moved towards) now have shorter lines, and A and D longer ones. Push the round far enough that one side of the round lines up with a side of the base and that side becomes vertical: the calculator's flat-side presets set this up. Open the offset example to see its pattern.

An offset round-to-round is the eccentric reducer, triangulated the same way. A centred round-to-round is a cone, which has a closed-form pattern: see the cone flat pattern formula.