Miter Angle Calculator
Calculate the miter angle to cut on each corner of a regular polygon frame, such as a picture frame, octagon, or decorative ring. Use it whenever you need every joint to meet flush without gaps.
About this calculator
For a regular polygon (all sides equal, all interior angles equal), each corner is formed by two mitered ends meeting at the interior angle. The miter angle you set on your saw is measured from square (90°), not from the cut face. The formula is: Miter Angle = 90 − (180 / Number of Sides). This derives from the fact that each corner's full interior angle is 180 − (360 / n) degrees, and each mating piece contributes exactly half of that angle. For a square (4 sides): 90 − (180/4) = 90 − 45 = 45°, which matches the familiar 45° miter for picture frames. For a hexagon (6 sides): 90 − (180/6) = 90 − 30 = 60°. The formula works for any regular polygon with 3 or more sides.
How to use
Suppose you want to build a regular octagon (8-sided) picture frame. Enter 8 as the number of sides. Calculate: Miter Angle = 90 − (180 / 8) = 90 − 22.5 = 67.5°. Set your miter saw to 67.5° from the fence (or equivalently, 22.5° from square, depending on how your saw's scale is oriented). Cut both ends of each of the 8 pieces at this angle, making sure the cuts are mirror images of each other. Dry-fit all pieces before gluing to confirm the joints close cleanly, then glue and clamp with band clamps or strap clamps designed for polygon assemblies.
Frequently asked questions
How do I convert the miter angle calculator result to a saw setting?
Most miter saws display the angle from square (0° = a 90° cut through the board). If the calculator returns 67.5°, subtract from 90° to get the saw setting: 90 − 67.5 = 22.5°, meaning you tilt the saw 22.5° from its default position. Some saws display the complementary angle directly. Always make a test cut on scrap first — set two pieces at the calculated angle and check that they meet flush with no gaps before cutting your finished material. Small errors in saw calibration compound around all n corners, so getting the angle right is critical.
What miter angle do I need for a hexagon, pentagon, or other common polygon?
Using the formula Miter Angle = 90 − (180 / n): a triangle (3 sides) requires 30°, a square (4 sides) 45°, a pentagon (5 sides) 54°, a hexagon (6 sides) 60°, an octagon (8 sides) 67.5°, and a dodecagon (12 sides) 75°. Notice that as the number of sides increases, the miter angle approaches 90°, meaning the cuts become nearly parallel to the board's length. For very high polygon counts used in decorative rings or coopered panels, even small angular errors become significant because they are multiplied across many joints.
Why do miter joints in polygon frames sometimes leave gaps even when cut at the correct angle?
Several factors cause gaps despite a correct theoretical angle. First, saw calibration drift means the blade is not precisely at the set angle — use a digital angle gauge to verify. Second, if the stock is not perfectly straight, the pieces rock during cutting and produce slightly different effective angles along the length. Third, cumulative error matters: in an octagon, an error of just 0.5° per joint produces a total angular error of 4°, which is clearly visible. Finally, measuring and cutting all pieces to exactly the same length is equally important, since length variation creates gaps even when angles are perfect. Dry-fitting and using a strap clamp to pull joints tight before measuring error points are good diagnostic steps.