How to use the segmented ring calculator
- Choose the units, then enter the outside diameter, measured across the flats of the finished ring, from the middle of one flat side to the middle of the opposite one.
- Enter the number of segments, the wall width (the width of the board, which sets the ring wall) and your saw kerf.
- Read the miter angle, the long and short edges of each segment and the board length needed for one ring.
- Use the cut list for the pieces. The second table shows the same ring with other segment counts.
Formula
miter angle at each end = 180 / n degrees
long edge = outside diameter x tan(180 / n)
short edge = (outside diameter - 2 x wall width) x tan(180 / n)
board length = n x ((long edge + short edge) / 2 + kerf / cos(180 / n))
diameter across the corners = diameter across the flats / cos(180 / n)
The segments are cut with every other one turned end for end, so each takes up the average of the long and short edges along the board. The kerf is measured square to the saw, so along an angled cut it takes up kerf / cos(angle).
A ring of 12 segments
A ring with an outside diameter of 10 inches, 12 segments, a wall width of 1.5 inches and a kerf of 0.125 inch has a miter of 15 degrees. The long edge is 10 x tan 15 degrees = 2.6795 inches and the short edge is 7 x tan 15 degrees = 1.8756 inches. The average is 2.2776 inches and the kerf along each cut is 0.1294 inch, so the board length is 12 x (2.2776 + 0.1294) = 28.8837 inches. Across the corners the ring measures 10.3528 inches.
A ring of 8 segments with an outside diameter of 200 mm, a wall width of 30 mm and a 3 mm kerf has a miter of 22.5 degrees, a long edge of 82.84 mm and a board length of 589.31 mm.
How segmented rings work
A segmented ring is a polygon cut into a closed band. Every segment is a trapezoid whose end angles are half of the angle that one segment spans around the ring. Stacking rings of different diameters builds a bowl or a vase, and staggering the joints from one ring to the next makes the stack strong.
A small error in a miter angle adds up around the ring, so cut a test ring and check it dry before you glue. With 12 segments, an error of half a degree at each end leaves a gap of several degrees around the circle. The miter angle calculator covers frames and corners with any number of sides, and the regular polygon properties calculator gives the sizes of the polygon itself.
Buying the wood
Board length is the length of one strip, so a ring that needs 28.9 inches can come from a board of that length and the width of the wall. Use the board foot calculator to turn lengths into volume, and the nominal to actual lumber converter to see the real size of store lumber. Wear eye and hearing protection, and follow your saw maker's instructions.
Assumptions and limits
- The outside of the ring is a regular polygon, and the diameter is measured across its flats.
- The inside is the same polygon with a diameter smaller by twice the wall width.
- The count of cuts equals the number of segments, including the cut that trims the first end. Add a little for squaring ends and test cuts.
- Between 3 and 60 segments are accepted.
Frequently asked questions
What is the difference between across the flats and across the corners?
Across the flats is the distance between opposite flat sides, which is the diameter of the circle that touches the sides. Across the corners is the longer distance between opposite points. The corner figure tells you the size of the blank you need to turn.
Why does the board length include the kerf?
Each saw cut removes a strip of wood, so the pieces together are shorter than the board. Adding the kerf for every cut keeps you from running short.
How many segments should I use?
More segments make a rounder ring with smaller miters but more cuts. Twelve is a popular choice because the miter is a convenient 15 degrees; sixteen, eighteen and twenty-four also work.
Can I use thickness here?
The ring thickness (its height) does not change the edges or the length. It sets the volume of wood, which you can total with a board foot tool.