Miter Angles, Circles and Arcs for Woodworkers
The geometry behind frames, polygons, round tops and curved parts
Woodworking hides a surprising amount of geometry. Whether you are making a picture frame, a six-sided planter, a round tabletop or an arched rail, the same few formulas appear again and again. The Angles and Circles tabs on the Woodworking Calculator apply them, and this page explains what the numbers mean so you can use them confidently at the saw.
Miter angles for regular polygons
A regular polygon with n sides has interior angles of (n − 2) × 180 ÷ n degrees. Two mitered boards meet at each corner, and each takes half of that angle. The calculator reports the interior angle and "each miter cut", which is half of the interior angle.
| Sides | Interior angle | Each miter (half of interior) | Saw setting from square |
|---|---|---|---|
| 3 | 60° | 30° | 60° |
| 4 | 90° | 45° | 45° |
| 5 | 108° | 54° | 36° |
| 6 | 120° | 60° | 30° |
| 8 | 135° | 67.5° | 22.5° |
| 10 | 144° | 72° | 18° |
| 12 | 150° | 75° | 15° |
Read this before you set the saw. The "each miter" figure is the angle between the cut face and the long edge of the board. Most miter saws and table saw gauges are marked with 0° as a square, straight-across cut, so the number to dial in is the last column: 90° minus the calculator's figure, which is the same as 180° divided by the number of sides. A square frame is 45° either way, but a hexagon needs 30° on the saw, not 60°. Always test on scrap, because a small angle error multiplies around the frame.
Compound angles
When the parts of a project lean as well as turn, such as splayed legs, a hopper or a faceted bowl, a single cut needs both a miter and a bevel setting. The Compound Angle Calculator takes a miter angle and a bevel angle and returns the saw miter and saw bevel settings using trigonometry: the saw miter is atan(tan(miter) × cos(bevel)) and the saw bevel is atan(tan(bevel) × sin(miter)).
Which angles you enter depends on the geometry of the piece, and there are several conventions for measuring them, so check how your plans define them. Cut a test piece, assemble it and confirm the fit with a square or bevel gauge before cutting the good stock.
Circles
For a round piece, enter the radius, which is half the diameter. The calculator returns:
- Diameter = 2 × r
- Circumference = 2 × π × r
- Area = π × r²
For a round tabletop 24 inches across, r = 12. The circumference is about 75.4 inches and the area about 452.4 square inches, or about 3.14 square feet. The circumference tells you how much edge banding or trim you will need; the area helps with finish quantities.
When cutting a circle with a router on a trammel or circle-cutting jig, the setting is the radius, measured from the pivot to the cutting edge of the bit. Remember to allow for the width of the bit if you measure to its center.
Arcs and chords
An arched rail, a curved apron or a bent lamination is a segment of a circle. Given the radius and the angle that the arc spans, the Arc & Chord calculator returns three numbers:
| Result | Formula | What it is for |
|---|---|---|
| Arc length | r × angle in radians | Length of the curved edge, for trim or the unrolled length of a bent part |
| Chord length | 2 × r × sin(angle ÷ 2) | The straight-line distance between the arc's ends, i.e. the width of the opening |
| Segment height | r × (1 − cos(angle ÷ 2)) | How far the curve rises above the chord, the "crown" |
For example, a 90° arc with a 24-inch radius has an arc length of about 37.7 inches, a chord of about 33.9 inches and a rise of about 7.0 inches. That is the shape of a gentle arch spanning roughly 34 inches.
Often you know the span and the rise rather than the radius. A useful relationship is radius = (chord² ÷ (8 × rise)) + (rise ÷ 2). You can enter the radius it gives into the calculator to check the arc length.
Unit conversions and fractions
The Conversions tab converts between inches, feet, millimeters, centimeters and meters, and turns fractions into decimals (and back, if you pick the Fractions format in the settings). It is handy for checking a plan that mixes metric and imperial measurements, or for entering a 5/8-inch figure into a calculator that wants decimals: 5 ÷ 8 = 0.625.
For the materials side of a project, see how to calculate board feet and cut lists and waste.