Sheet metal
Square to round calculator
In short
A square to round is developed by triangulation: four flat triangles on the rectangle sides and four corner fans to the circle, divided into N = 4m chords of length c = 2R sin(π / N). Each element line is laid out at its true length ℓ = √(Δx² + Δy² + h²), as the plan length is not enough.
Calculation
Example calculated with the tool’s default values.
Inputs
- Rectangle width (a)
- 400 mm
- Rectangle depth (b)
- 300 mm
- Round diameter (Ø)
- 200 mm
- Height between the two openings
- 250 mm
- Circle divisions per quarter
- 3
- Template
- Automatic (half if symmetric)
Results
- Minimum plate for the template
- 487 × 453 mmwidth × height of the half pattern
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Longest element line
- 324.0 mm
- Shortest element line
- 292.2 mm
- Circle chord
- 51.76 mm12 chords
- Developed area
- 0.2703 m²
- Weld seam length
- 269.3 mmcut P0–M0 (a single weld for the complete piece)
- Rectangle perimeter
- 1400.0 mm
- Dimensions are taken on the neutral fibre (mid-thickness). With outside or inside dimensions, subtract or add the thickness before entering them.
- The circle is laid out with 12 chords: its developed perimeter is 621.2 mm, 1.14 % below π × Ø = 628.3 mm. The finished round is formed on the rolls or press between the points.
- Half pattern: lay it out twice (or flip it); each straight axis edge is half a side, to be welded between pieces.
- The four corner edges are press-bent; the fans are rolled. Check the layout before cutting: the circle must close on N equal chords.
Step-by-step calculation
Circle points
N = 4m ; φi = i × 360° / N ; Pi = (ex + R cos φi ; ey + R sin φi ; h)
N = 4 × 3 = 12; pitch 30.00°; R = 100.0 mm
Circle chord
c = 2R sin(π / N)
2 × 100.0 × sin 15.00° = 51.76 mm
True length of an element line
ℓ = √(Δx² + Δy² + h²)
A–P0 : √(-100.0² + -150.0² + 250.0²) = 308.22 mm
Flat triangles and fans
4 triangles (sides) + 4 fans of m triangles (corners)
4 + 4 × 3 = 16 triangles
Unfolding
each triangle laid on the shared edge: two compass arcs of radii ℓ1 and ℓ2
9 triangles unfolded in the order M0 → A → fan A → …; seam P0–M0 = 269.3 mm
Surface area
S = Σ areas of the triangles in space
0.2703 m²
True lengths of the element lines from corner A (B, C, D identical by symmetry)
| Corner | Circle point | Plan length (mm) | True length (mm) |
|---|---|---|---|
| A | 0 | 180.3 | 308.2 |
| A | 1 | 151.2 | 292.2 |
| A | 2 | 162.8 | 298.4 |
| A | 3 | 206.2 | 324.0 |
Check lengths (the layout must reproduce them)
| Item | Length (mm) |
|---|---|
| Circle chord (12 chords) | 51.76 |
| Side A–B (= C–D) | 400.00 |
| Side B–C (= D–A) | 300.00 |
| Seam P0–M0 | 269.26 |
How to use it
- Enter the rectangle width a and depth b, the round diameter and the height between the two openings.
- Pick the number of circle divisions per quarter (3 per quarter = 12 equal parts).
- Enter the round offsets ex and ey for an offset transition, or leave them empty if it is centred.
- Pick the part to lay out: automatic, complete piece, half or quarter.
- Read the minimum plate size (width × height) of the pattern.
Frequently asked questions
How do I lay out a square to round?
By triangulation: divide the circle into N = 4m points, join each rectangle corner to the m + 1 circle points of its quarter (conical fan) and each side to the circle point facing it (flat triangle). Work out the true length of each element line, then lay the triangles flat one after the other, each with two compass arcs of radii ℓ1 and ℓ2, starting from a seam P0–M0.
How are the true lengths of a square to round development found?
ℓ = √(Δx² + Δy² + h²), with Δx and Δy the plan offsets between the corner and the circle point and h the height between the openings. Example: 400 × 300 rectangle, Ø 200 round, h = 250: A–P0 = √(-100.0² + -150.0² + 250.0²) = 308.22 mm. The calculation also gives the longest and shortest element lines, which the layout must reproduce.
What does a worked square to round example give?
With a 400 × 300 rectangle, a Ø 200 round, a height of 250 and 3 divisions per quarter: N = 4 × 3 = 12, chord c = 2 × 100.0 × sin 15.00° = 51.76 mm, 16 triangles, developed area 0.2703 m², P0–M0 weld seam of 269.3 mm, and a half pattern that fits on a 487 × 453 mm plate.
Can an offset square to round be laid out?
Yes: enter the round offsets ex and ey from the rectangle centre. The pattern is then calculated as a complete piece; a half pattern needs a centred round and a quarter a centred square (a = b). If the round overhangs the rectangle in plan, the surface is strongly twisted: check the fit-up before cutting.
Limits
Indicative results. Dimensions are taken on the neutral fibre: with outside or inside dimensions, subtract or add the thickness before entering them. The circle is laid out with chords: with 12 chords, the developed perimeter is 1.14 % below π × Ø; the finished round is formed on the rolls or press. Check the layout before cutting.
Sources
- Transition carré-rond centrée : développement par triangulation, lignes d’élément, vraies grandeurs : Pattern Development: Sheet Metal Level 1 (Open Textbook BC / eCampusOntario), chap. 24 « On-Centre Square to Round » (link)
- Quatre triangles plans et quatre fuseaux de coin ; vraie grandeur ℓ = √(h² + Δx² + Δy²) ; cercle en 12 cordes : Planichapa, « Square to round transition: development by triangulation » (consulté le 08/11/2026) (link)
- Géométrie : Distance entre deux points de l’espace ; corde 2R sin(π/N) ; construction d’un triangle à partir de ses trois côtés