How to lay out a square-to-round by hand

A square-to-round is the classic exercise in triangulation: four flat triangles and four corners, each corner laid out as a fan of lines from the rectangle corner to the round. Here it is step by step on a worked example — with one correction to the book method that keeps the round edge from coming out short.

Published 2026-10-30. Rather have the numbers worked out? Square / Rectangle to Round Transition does it in your browser.

What you are making

The worked example: a 600 × 400 rectangle at the bottom, a 300 round at the top, 350 apart, the round centred over the rectangle, with the seam up the middle of the front side — where most of the books put it. For plate, work to the mid-thickness dimensions.

The surface has two kinds of piece. On each side of the rectangle there is a flat triangle whose tip, the apex, is the point of the round whose tangent runs parallel to that side — on a centred part like this one, straight in from the middle of the side. Between them, each corner joins one rectangle corner to a quarter of the round: a piece of an oblique cone, laid out as a fan of lines (Kittredge, Problem 177).

Figure 1. Plan. Each quarter of the round divided into four, joined to its corner; the front-right corner's points are numbered 0 (the front apex) to 4 (the right apex).

1. Divide the round

Divide each quarter of the round into equal parts and join every division to the rectangle corner it belongs to. Kidder uses four parts per quarter, sixteen round the circle; Coey uses three. More parts give a smoother round edge.

2. Find the true lengths

None of these lines shows its true length in plan or elevation, because each one leans in two directions. Find each from a right triangle: one side is the height, 350; the other is the line's length in plan. The front side's slant, from the middle of the front side (0, −200) to the front apex (0, −150), is √(50² + 350²) = 353.55; from the corner (300, −200) to the same apex it is √(300² + 50² + 350²) = 463.68.

Side triangles
Side	Length	Slant (middle → apex)	Corner → apex
Front and back	600.00	353.55	463.68
Right and left	400.00	380.79	430.12

And the fan across the front-right corner — the other three corners are mirror images on a centred part:

Front-right corner fan
Line	Corner → round (true length)	Step to the next point, on the pattern
0	463.68	58.87
1	430.26	58.82
2	411.02	58.81
3	410.96	58.82
4	430.12	—

3. Lay out the back triangle

Start in the middle of the sheet with a whole side. Draw the back side, 600.00 long. From each end swing 463.68; where the arcs cross is the back apex. Check: from the middle of the side to the apex measures 353.55.

4. Fan round the corner

From the back-right corner swing the true length of the next fan line; from the apex you are at, swing the step; where they cross is the next point of the round edge. Repeat until you reach the last line of the fan — that point is the right apex.

5. The side triangle, then the next corner

From the back-right corner swing the side length 400.00, and from the right apex the corner-to-apex length 430.12: the crossing is the front-right corner. Fan round the front-right corner to the front apex the same way. You arrive at its last line, so read that table from the bottom up — line 4 back to line 0 — taking the step between two lines from the upper of the two rows.

6. Finish at the seam

The seam runs from the middle of the front side to the front apex. From the front-right corner swing half the front side, 300.00, and from the front apex the seam length, 353.55: the crossing is the foot of the seam. On a centred part the seam meets the side square, which is a useful check. Do the left half the same way from the back triangle — or mirror the half you have.

7. Join and check

Join the points of the round edge with a smooth curve and the rectangle corners with straight lines. The rectangle edge must measure the perimeter, 2 × (600 + 400) = 2000; the round edge π × 300 = 942.48; both seam lines 353.55.

Figure 2. The finished pattern from outside: the rectangle edge round the outside, the round edge on the inside, the corner fans dashed in green and the edges of the flat triangles in grey. It fits a blank about 749.62 × 1327.31.

In two pieces

A transition too large for one sheet is made in pieces; the ICS textbook says developing only a half or a quarter is customary for that reason (§18, p. 55). Where the books draw a half pattern, most put its seams at the middles of two opposite sides (Kittredge, Problem 176; Longfield, Problem 46), and Daugherty’s chimney base is made that way, in two pieces (Problem 107). They mostly keep a long side whole; here the cut is at the front and back, to match the one-piece seam. For one half, begin step 3 with only half the back side, 300.00: from the middle of the back side swing the slant, 353.55, and from the back-right corner the corner-to-apex length, 463.68; the crossing is the back apex, and on this centred part the slant stands square to the side. Then carry on with steps 4 to 6, stopping at the foot of the front seam. That half — half the back side, the right side with its two corners, half the front — is one piece, with a seam at each end, each 353.55 long. On a centred part the other half is the same shape. The tool lays each half out with the two ends of its rectangle edge level, in a blank about 869.99 × 463.43. Add laps or locks on both seam edges (Daugherty; Longfield).

With the round offset the halves differ. Lay each one out from its own true lengths, unless the round is offset along the line the part is split on, when the second half is the first turned over (Kidder, pp. 26–28). No book we found works a square-to-round through in four pieces, though ICS calls developing a quarter customary for large ones and the books draft the quarter between two mid-side lines; the tool will split it there.

The one correction: step on the pattern, not the plan chord

The books step the spacing of the round straight from the plan — here the chord 58.53. A chord is always shorter than its arc, as Kidder points out (p. 66), so four chords give 234.11 per quarter against 235.33 for the four steps measured on the flat pattern: stepping chords lands about 4.88 short round the whole edge, a gap you see at the seam. The steps in the table above are measured on the exact flat pattern — 58.87, 58.82, … — and four of them make 235.33, with the smooth curve through the points the full 235.62. Kidder's own remedy is to divide the computed circumference instead of stepping chords; the pattern steps do the same job exactly. The tool prints a fan table like this for every corner, offset parts included.

Common mistakes

Using plan lengths as if they were true lengths — every line here leans.
Putting the apex of a side triangle anywhere but the point of the round whose tangent is parallel to that side — on an offset part that is not straight in from the middle.
Fanning a corner in the wrong direction and ending up with the outside face on the inside.
Stepping the plan chord round the corners and finding the round edge short.
On an offset part cut in two, using the first half as the template for the second: the halves differ, unless the round is offset along the split line and the template is turned over.

How this matches the tool

The square-to-round tool develops each corner exactly, by numerical integration, and a separate program checks it by triangulating with over four thousand triangles per corner. Laid out by hand with the fan table's steps, your points sit on that same outline. Every number on this page was produced by the tool's own code when the page was built.

Sources

Skip the arithmetic

Square / Rectangle to Round Transition — One exact flat pattern for a square-to-round or rectangle-to-round, centred or offset — with true lengths, corner bend lines and a DXF.

Free, in your browser; nothing you type is sent anywhere. How the tools are checked is on the Verification page. Rather have the part drawn for you? Drafting for small shops, quoted per job.