How to lay out a square to round
To lay out a square to round, divide each quarter of the round into equal spaces and find the true length from each corner to every point: the square root of height squared plus plan distance squared. From the seam, swing each point from the last two, one triangle at a time, and join the round with a curve.
Drawn by Stretchout from the worked example below. Triangulation development.
What you need
- Sheet or plate for the blank
- A tape and dividers
- A straightedge
- A flexible batten for the round edge
- A brake for the corner bends
Worked example
We’ll lay out a 24" × 24" square to a 14" round, 16" high, centered, in 3/16" plate, one piece, 4 spaces per corner, laid out on the mean line. The numbers:
- Square, on the mean line
- 23 13/16" × 23 13/16"
- Round, on the mean line
- 13 13/16" across
- Height
- 16"
- Corner lines, true length
- 20 9/16", 19 5/16", 18 13/16", 19 5/16", 20 9/16"
- Seam
- 16 3/4"
- Blank
- 63 3/16" × 29 3/4"
Step by step
Step 1
Draw the first seam, 16 3/4" long. Its end on the square end is point 1, on the round point a.
Step 2
From 1 swing 11 15/16" along the square end. From a swing 20 9/16". Where they cross is point 2.
Step 3
From a swing 2 11/16" along the round. From 2 swing 19 5/16". Where they cross is point b.
Step 4
Keep going the same way, one point at a time. Each new point is where the two swings cross:
Point Swing from and swing from c 2 11/16" b 18 13/16" 2 d 2 11/16" c 19 5/16" 2 e 2 11/16" d 20 9/16" 2 3 23 13/16" 2 20 9/16" e f 2 11/16" e 19 5/16" 3 g 2 11/16" f 18 13/16" 3 h 2 11/16" g 19 5/16" 3 i 2 11/16" h 20 9/16" 3 4 23 13/16" 3 20 9/16" i j 2 11/16" i 19 5/16" 4 k 2 11/16" j 18 13/16" 4 l 2 11/16" k 19 5/16" 4 m 2 11/16" l 20 9/16" 4 5 23 13/16" 4 20 9/16" m n 2 11/16" m 19 5/16" 5 o 2 11/16" n 18 13/16" 5 p 2 11/16" o 19 5/16" 5 q 2 11/16" p 20 9/16" 5 6 11 15/16" 5 16 3/4" q Step 5
Join the points: straight across the square end, a smooth curve through the round. Cut on it and the seams. Bend on the corner lines.
Watch it laid out
Every step of the worked example. Play, or step through with Next and Back (or the arrow keys).
1.Draw the first seam, 16 3/4" long. Its end on the square end is point 1, on the round point a.
The true lengths
Take a and b as half the square’s sides, r as the round’s radius, and H as the height, all on the line you lay out on. From a corner at (a, b), the true length to the point at angle φ on the round is √(H² + (a − r·cos φ)² + (b − r·sin φ)²).
For a centered square, each corner fans across a quarter of the round, φ from 0° to 90°, and all four corners have the same lengths. The shortest line from each corner sits in the middle of its fan.
The flat triangles between the fans are flat sides of the square end. A one-piece pattern puts the seam in the middle of one of them, so the seam is a straight line from the middle of that side up to the round.
Questions
Does this work for a rectangle to round?
Yes. Use the two half sizes a and b; the corners just fan across different parts of the round. The calculator takes any width and depth.
What if the round is off center?
Every corner gets its own set of lengths, but the method is the same. Switch on Offset in the calculator and it lays out each corner.
What about a pitched (sloped) end?
When the round sits at an angle, the lengths change all the way around. The calculator’s Pitch option solves it, including where the flat sides meet the round.
How many pieces should it be?
One piece for small ones, two or four for big ones or to fit the sheet. Seams go in the middle of the flat sides.
Where do I bend it?
Along the lines from each corner at the ends of its fan: those are the brake lines. The fans roll; the flat triangles stay flat.
Your sizes
Below is the square to round calculator, opened on the worked example. Put in your own sizes and it redoes every number and every step for your fitting. It’s also at /stretchout/shapes/square-to-round.