What this calculator does
Find the exact angle to set your miter saw so the corners of a frame or box close tight. A miter joint cuts both pieces at an angle, so they meet along a line that splits the corner in two. That hides the rough end grain. For a square picture frame, the angle is the familiar 45°. But a hexagonal planter, an eight-sided table apron or a frame around an odd corner needs a different setting. Getting it slightly wrong opens a gap at every joint.
This calculator works out the saw setting from the number of sides of an even-sided shape. Or it works from any corner angle you measure. Sometimes the sides lean outward, as on a sloped box, a hopper or a tapered planter. Then each piece also needs the blade tilted, which is called a compound miter. The calculator gives the miter and the bevel (blade tilt) together. Saw scales differ, so check the reading on a test cut in scrap before cutting good wood.
Where it's used
Why this math matters outside the classroom:
- Picture frames. Framers cut every corner of a rectangular frame at 45°, because each piece takes half of the 90° corner.
- Hexagonal planter. A garden builder making a six-sided planter sets the saw to 180 ÷ 6 = 30° for every cut.
- Octagon gazebo rails. Carpenters fitting railings between the posts of an octagonal gazebo cut 22.5° miters on each end.
- Bay window trim. Trim carpenters casing a 135° bay window corner set the saw to 90 − 67.5 = 22.5° for each piece.
- Sloped hoppers. Woodworkers building a four-sided hopper with sides leaning 15° cut a compound joint of about a 14.5° miter and 43.1° bevel.
Grade level and standards: Grades 7–9; Common Core State Standards for Mathematics 7.G.B.5, HSG.SRT.C.8. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
How to use it
- Choose whether you know the number of sides or the corner angle.
- Enter the number of sides (4 for a frame, 6 for a hexagon) or the corner angle in degrees.
- If the sides lean outward, tick the box. Then enter how far each side leans from straight up.
- Read the miter setting, plus the bevel for leaning sides. Make a test cut in scrap and check the fit.
What the inputs mean
- Number of sides
- Every side must be the same length, and every corner the same angle, for one saw setting to fit all the joints.
- Corner angle between the two pieces
- The angle the finished pieces make with each other, measured on the inside of the joint. A square corner is 90°, and a wall that turns a bay window is often 135°. A bevel gauge copies an odd corner so you can read it on a protractor.
- Side lean from vertical
- How far each side tips away from upright. 0° is a box with straight-up sides, which needs a bevel only. A frame lying flat on a table is the other extreme and needs a miter only.
Formula and method
corner angle of a regular polygon = 180° − 360° ÷ n flat frame: miter setting = 90° − corner ÷ 2 (= 180° ÷ n for a regular polygon) leaning sides: miter = arctan(sin lean ÷ tan(corner ÷ 2)) leaning sides: bevel = arcsin(cos lean × cos(corner ÷ 2))
Here n is the number of sides, and "corner" is the inside angle where two pieces meet. "Lean" is how far each side tips from straight up. A regular polygon is a shape whose sides and corners are all equal. In a flat frame, each of the two pieces at a joint takes exactly half of the corner. A miter saw's scale reads 0° for a square cut, so the setting is 90° minus that half angle. That's 45° for a square corner, 30° for a hexagon and 22.5° for an octagon.
When the sides lean, the joint line runs at a slant through the corner, and the cut becomes a compound miter. The two formulas are the standard ones woodworkers use for it. Sin, cos and tan are the usual triangle functions on a scientific calculator, and arctan and arcsin turn their results back into angles. With a 90° lean (sides lying flat), they match the flat-frame setting. With a 0° lean (upright sides), they give a pure bevel of 90° minus half the corner.
Worked example
You're building an eight-sided frame that lies flat, with all sides equal.
- Corner angle: 180° − 360° ÷ 8 = 135°.
- Half the corner: 135° ÷ 2 = 67.5°. That's the angle between each cut and the edge of the board.
- Miter setting: 90° − 67.5° = 22.5°.
Result: Set the miter saw to 22.5° with the blade straight up. Sixteen of these cuts close the octagon.
Understanding the result
You read the miter setting on the saw's miter scale, where 0° is a square cut. The angle between the cut line and the board's edge is what's left of 90°. It's shown separately, for marking with a protractor or bevel gauge.
For leaning sides, set both the miter and the bevel before cutting. Flip the board face up and face down between cuts, or swing the saw the other way. That gives matching pieces mirror-image cuts.
Assumptions and limitations
- All pieces in a shape are the same width and thickness, and every corner has the same angle.
- Both pieces of a joint lean by the same amount.
- The saw's scales are accurate. Small errors multiply over many joints.
- These settings are ideal ones. Real wood, a blade that flexes and slightly-off scales mean a test cut and small tweaks are part of the job.
- Pieces of different widths or leans, such as a sloped side meeting an upright one, need a separate calculation for each piece.
- Crown molding uses the same compound angles, with the spring angle as the lean. The crown molding calculator handles its own conventions.
Frequently asked questions
What angle do I cut for a hexagon or an octagon?
Divide 180° by the number of sides. A hexagon needs 30° on the miter scale, and an octagon 22.5°. A pentagon needs 36°, and a twelve-sided shape 15°. Cut each piece at that setting on both ends, angled in opposite directions.
Why does my 45° frame still have gaps?
Small errors add up. A frame has eight cut faces, so a setting off by half a degree adds up to 4° around the frame. That error shows up as an open joint. Check the saw with a test frame, and make sure opposite sides are exactly the same length. Then adjust in small steps.
How do I cut the corners of a box with sloped sides?
Tick the leaning-sides box and enter the lean from straight up. A four-sided box with sides leaning 15° needs about a 14.51° miter and a 43.08° bevel. The more the sides lean, the more the cut shifts from a bevel toward a plain miter.
What is the difference between the miter and the bevel?
The miter is the angle the saw swings across the board, seen from above. The bevel is how far the blade tilts away from straight up. A flat frame needs only a miter, and an upright box corner only a bevel. Leaning sides need both at once.
How this calculator is checked
Method: Polygon interior angle (180 − 360/n) and the standard compound-miter formulas for tilted faces: miter = arctan(sin φ / tan h), bevel = arcsin(cos φ · cos h). The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.
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