Miter Angle Calculator for Polygons and Rings
Choose the number of sides and the size of the outside.
Your measurements
Enter the number of sides and the size of the outside. Everything updates as you type.
The shape
4 for a picture frame, 6 for a hexagon, 8 for an octagon. Rings are often 8 to 16.
The distance between two opposite sides, measured square to them.
Measured on the outside of the ring or frame.
The board and the saw
The width of the board the segments are cut from. Leave at 0 for a plain polygon.
The width of the cut the blade makes. Many saw blades are about 1/8 in (3.2 mm). Leave at 0 to ignore it.
Results
Miter saw angle for each cut
°
= from a square cut
8 sides: two cuts of 22.5° meet to make a 135° corner. Use the same angle for every cut and flip the board between cuts.
- Long edge
- 4 31/32"
- = 126.3 mm
- outside of each segment
- Short edge
- 3 23/32"
- = 94.7 mm
- inside of each segment
- Board length
- 35 7/8"
- = 911.3 mm
- 8 segments, with kerf
- Corner angle
- 135°
- 45° at the centre
Size of the polygon
Fractions are to the nearest 1/32 in. The long edge is the outside edge.
Size of the polygon
Fractions are to the nearest 1/32 in. The long edge is the outside edge.
| Measurement | Inches | Millimetres |
|---|---|---|
| Edge (side) | 4.9706 | 126.3 |
| Across flats | 12 | 304.8 |
| Across corners | 12.9887 | 329.9 |
| Apothem (centre to a side) | 6 | 152.4 |
| Circumradius (centre to a corner) | 6.4944 | 165 |
| Kerf along the board at the angle | 0.1353 | 3.44 |
How this was calculated
Angle, edges and board length, with your numbers.
How this was calculated
Angle, edges and board length, with your numbers.
Step 1: Saw angle for each cut
miter angle = 180 ÷ N
= 180 ÷ 8
= 22.5° from squareTwo cuts of 22.5° meet to make the 135° corner of the polygon (180 − 360 ÷ N). For a ring the saw is set to 22.5° and the board lies flat on the table.
Step 2: Edge length from across flats
edge = across flats × tan(180 ÷ N)
= 12 × tan(22.5°) = 12 × 0.414214
= 4.9706 inAcross flats is twice the apothem, the distance from the centre to the middle of a side. For a polygon, side = 2 × apothem × tan(180 ÷ N).
Step 3: Long and short edge of each segment
short edge = long edge − 2 × board width × tan(180 ÷ N)
= 4.9706 − 2 × 1.5 × 0.414214
= long 4.9706 in, short 3.7279 inEach end of the board is cut at the miter angle, so each end loses board width × tan(angle) on the inside edge.
Step 4: Board length needed
N × ((long + short) ÷ 2 + kerf ÷ cos(180 ÷ N))
= 8 × ((4.9706 + 3.7279) ÷ 2 + 0.125 ÷ cos(22.5°))
= 35.8763 inCutting the segments from one board, flipping it between cuts, each segment uses the average of its long and short edge. An angled cut wastes a little more than the kerf: kerf ÷ cos(angle). Add length for trimming the ends and for defects.
In short
Miter angle = 180 ÷ N. An 8-sided ring 12 in across flats, cut from a 1.5 in board, needs 22.5° cuts. Each segment is 4.9706 in on the long edge and 3.7279 in on the short edge (126.3 mm and 94.7 mm).
The calculator gives the saw angle, the long and short edge of each trapezoid segment and the board length needed.
Miter saw angles for 3 to 12 sides
The angle depends only on the number of sides: 180 ÷ N from a square cut. Angles are rounded to two decimals.
| Sides | Miter saw angle | Corner angle | Where it is used |
|---|---|---|---|
| 3 | 60° | 60° | Triangle: a triangular frame or shelf |
| 4 | 45° | 90° | Picture frame or square box (45 degree miters) |
| 5 | 36° | 108° | Pentagon box or planter |
| 6 | 30° | 120° | Hexagon planter, table top or shelf |
| 7 | 25.71° | 128.57° | Seven-sided design (uncommon) |
| 8 | 22.5° | 135° | Octagon: frame, gazebo roof, segmented ring |
| 9 | 20° | 140° | Nine-sided design (uncommon) |
| 10 | 18° | 144° | Segmented bowl or vase ring |
| 11 | 16.36° | 147.27° | Eleven-sided design (uncommon) |
| 12 | 15° | 150° | Segmented bowl ring or round table edge |
The angles are geometry and do not vary. The uses are examples, not rules.
Worked example: 8 sides, 12 in across flats, 1.5 in board
An octagonal ring with a 0.125 in saw kerf.
Step 1: Saw angle for each cut
miter angle = 180 ÷ N
= 180 ÷ 8
= 22.5° from squareTwo cuts of 22.5° meet to make the 135° corner of the polygon (180 − 360 ÷ N). For a ring the saw is set to 22.5° and the board lies flat on the table.
Step 2: Edge length from across flats
edge = across flats × tan(180 ÷ N)
= 12 × tan(22.5°) = 12 × 0.414214
= 4.9706 inAcross flats is twice the apothem, the distance from the centre to the middle of a side. For a polygon, side = 2 × apothem × tan(180 ÷ N).
Step 3: Long and short edge of each segment
short edge = long edge − 2 × board width × tan(180 ÷ N)
= 4.9706 − 2 × 1.5 × 0.414214
= long 4.9706 in, short 3.7279 inEach end of the board is cut at the miter angle, so each end loses board width × tan(angle) on the inside edge.
Step 4: Board length needed
N × ((long + short) ÷ 2 + kerf ÷ cos(180 ÷ N))
= 8 × ((4.9706 + 3.7279) ÷ 2 + 0.125 ÷ cos(22.5°))
= 35.8763 inCutting the segments from one board, flipping it between cuts, each segment uses the average of its long and short edge. An angled cut wastes a little more than the kerf: kerf ÷ cos(angle). Add length for trimming the ends and for defects.
Before you cut
- Make a test cut in scrap and check the corner with a square or a drafting angle. A small saw error of half a degree opens a gap over a ring of many segments.
- Some saws and sleds read the angle from the fence and some from square. The angle here is from a square cut, so 45 degrees makes a square corner.
- Measure the finished segment on its long edge. The long edge is the outside of the ring, and the short edge is the inside.
- The board length is a guide for flipping one board between cuts. Cut a spare segment or two for dry fitting.
Miter angles: common questions
Angles, segments and board length.
What miter saw angle do I use for a polygon?
Divide 180 by the number of sides. A four-sided frame is 180 ÷ 4 = 45 degrees, a hexagon is 30 degrees and an octagon is 22.5 degrees. Set the saw to that angle from a square cut and use it for every cut. Two cuts meet to make the corner angle, which is 180 − 360 ÷ N.
How long is each segment of a segmented ring?
Each segment is a trapezoid. Its long edge is the outside edge of the ring: 2 × apothem × tan(180 ÷ N), where the apothem is half the distance across flats. Its short edge is the long edge minus 2 × the board width × tan(180 ÷ N), because each end loses that much on the inside. The calculator shows both.
What is the difference between across flats and across corners?
Across flats is the distance between two opposite sides. Across corners is the distance between two opposite corners, and it is always the longer of the two. For a hexagon, across corners is about 15% more than across flats. With an odd number of sides no two sides are parallel, so the calculator reads these as twice the distance from the centre to a side or to a corner. Edge length avoids the question.
How much board do I need for a ring?
If you cut the segments from one board and flip it between cuts, each segment uses the average of its long and short edge, plus the saw kerf spread along the angled cut. The result is shown for all the segments together. It does not include trimming the ends of the board, defects, or spare segments, so add a little. A board wider than the apothem cannot make a ring at all, and the calculator says so.