Calculate regular polygon notching angles from the number of sides, or find the closest side count from a notch, exterior, or miter angle.
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Notching Angle Formula
Reverse formulas used by the calculator:
- n โ number of sides in the regular polygon
- Notching angle โ the inside angle of the corner you cut into the material
- Exterior angle โ the outside turn between two adjacent sides (360ยฐ / n)
- Miter setting โ the deflection from a square cut for each equal miter (exterior / 2); this is not the cut-line angle measured from the board edge
The formula assumes a regular polygon with equal sides and equal angles. For tube or bar notching where two pieces meet at a joint, the notching angle is the included angle between the two pieces. Convert radians to degrees with deg = rad ร 180 / ฯ before applying the side-count formulas.
Common Notching Angles by Shape
Use these as a quick reference when laying out a frame or a segmented ring.
| Sides | Shape | Notch angle | Miter setting |
|---|---|---|---|
| 3 | Triangle | 60ยฐ | 60ยฐ |
| 4 | Square | 90ยฐ | 45ยฐ |
| 5 | Pentagon | 108ยฐ | 36ยฐ |
| 6 | Hexagon | 120ยฐ | 30ยฐ |
| 8 | Octagon | 135ยฐ | 22.5ยฐ |
| 10 | Decagon | 144ยฐ | 18ยฐ |
| 12 | Dodecagon | 150ยฐ | 15ยฐ |
| 16 | Hexadecagon | 157.5ยฐ | 11.25ยฐ |
Which angle to enter on the saw or layout depends on how the tool is calibrated.
| Angle name | What it measures | Hexagon example |
|---|---|---|
| Notch / inside angle | Opening between two sides at the joint | 120ยฐ |
| Exterior angle | How much you turn at each corner | 60ยฐ |
| Miter setting | Saw angle for each of the two cuts | 30ยฐ |
| Cut from edge | Half the notch, measured off the side | 60ยฐ |
Worked Example
You want to build a hexagonal planter from six equal boards. Plug n = 6 into the formula:
Notching angle = 180ยฐ โ (360ยฐ / 6) = 180ยฐ โ 60ยฐ = 120ยฐ.
Each board needs a 30ยฐ miter cut on both ends, since 60ยฐ / 2 = 30ยฐ. The two miter cuts meet to form the 120ยฐ interior corner.
FAQ
Is the notching angle the same as the miter angle? No. The notch is the full inside angle. The miter setting is half of the exterior angle, because two cuts share the corner.
What if my saw measures from 0ยฐ as a square cut? Most miter saws read the deflection from 90ยฐ, which equals the miter setting in the table. Set a hexagon corner to 30ยฐ on that scale.
Can I use this for tube notching where two pipes meet? The inside-angle convention can describe a joint, but the inverse side-count calculation assumes a convex regular polygon. An arbitrary tube joint need not fit that model. Tube diameters and orientation determine the saddle cut profile, which this calculator does not calculate.
Why does the calculator round the side count? A regular polygon requires an integer side count. If the inferred count is 5.7, it rounds to 6 and reports that layoutโs angles and signed angle difference. The nearest integer side count does not always minimize angle error.
