Square-to-Round Transitions: How to Develop the Flat Pattern for Sheet Metal

 A square-to-round transition is one of those pieces that shows up everywhere in fabrication  custom exhaust collectors, HVAC plenums, hopper outlets, intake boxes  and it's also one of the first things that separates a fabricator who understands geometry from one who's just guessing with a hammer.

This post covers how to develop the flat pattern from scratch using the triangulation method. No software required, though I'll mention how CAD handles it at the end.

What You're Actually Making

A square-to-round transition connects a rectangular opening at one end to a circular opening at the other. The two openings are usually centered on the same axis (concentric) but they don't have to be  offset transitions exist too. This post covers the concentric version.

The shape is not a cone. It's not a pyramid. It's a hybrid surface made up of four flat triangular panels (the corner sections) and four curved triangular sections (the flat-to-round sides). Understanding that distinction is the whole key to laying it out correctly.

What You Need Before You Start

Before picking up a scribe, you need four dimensions:

1. Square end width (W)  e.g. 200mm

2. Square end length (L)  e.g. 200mm (square in this case)

3. Round end diameter (D)  e.g. 120mm

4. Height (H)  the vertical rise of the transition, e.g. 150mm

For this walkthrough: W = 200mm, L = 200mm, D = 120mm, H = 150mm. Work in whatever units you use  just stay consistent.

Step 1: Set Up Your Reference Points

Draw the square base centered at the origin. Label the four corners A, B, C, D going clockwise. Their coordinates:

A = (100, 100)

B = (100, -100)

C = (-100, -100)

D = (-100, 100)

Now draw the circle at the top, centered at (0, 0, 150). Divide the circle into 12 equal segments  30 degrees apart. Label these points 1 through 12. More divisions = smoother curve on your finished piece. 12 is a good balance between accuracy and the amount of layout work.

Each segment point on the circle is at:

x = (D/2) x cos(n x 30deg)

y = (D/2) x sin(n x 30deg)

z = H

For D = 120mm, radius = 60mm.

Step 2: Connect Corners to Circle Points (Triangulation)

This is the triangulation method. You're breaking the entire surface into a series of flat triangles that, when unfolded, give you the flat pattern.

For each quadrant of the transition (between two adjacent square corners), you connect:

- The square corner to the nearest circle points in that quadrant

- Adjacent circle points to each other

For the quadrant between corner A and corner B, the circle points that fall in that zone are points 1 through 4 (roughly the quarter of the circle closest to that side).

Draw triangles: A-1, A-1-2, A-2-3, A-3-B, and so on. You're meshing the surface with triangles.

Step 3: Calculate True Lengths

Here's the part most people skip and then wonder why their pattern doesn't fit.

Every line you drew in step 2 is a plan view line  it's the horizontal distance only. The true 3D length of each line is longer because it also rises by height H (or a portion of it).

True length formula:

TL = sqrt(plan_length^2 + height_difference^2)

For a line from corner A at z=0 to circle point 1 at z=150:

- Plan length = distance from A to point 1 in the XY plane

- Height difference = 150mm

- TL = sqrt(plan_length^2 + 150^2)

Do this for every triangle edge. Build a true length diagram  a simple right triangle construction on your layout sheet where the base is the plan length and the height is the z difference. The hypotenuse is your true length. Old school, fast, accurate.

Step 4: Develop the Flat Pattern by Triangulation

Now you unroll the surface triangle by triangle:

1. Start with a baseline  draw the true length of edge A-B (this is a flat edge, so it equals the actual side length: 200mm).

2. From A, swing an arc equal to the true length of A-1.

3. From B, swing an arc equal to the true length of B-4 (the nearest circle point on the B side).

4. Where those arcs intersect gives you point 1 and point 4 in your flat pattern.

5. Continue outward: from point 1, swing arc = chord length 1-2 (the arc between circle divisions, approximated as a straight chord). From A, swing arc = true length A-2. Intersection gives point 2.

6. Repeat for every triangle until the full quadrant is developed.

7. Mirror or repeat for all four sides.

The result is a flat shape you can cut from sheet and fold up into your transition. The four curved edges will form the round opening; the straig