Miter Angle Calculator for Woodworking & Construction

What Is a Miter Angle?

A miter angle is the angle a saw or other cutting tool is set to when cutting the ends of boards that will meet at a corner. In woodworking and construction, a miter cut creates a tight corner joint on picture frames, boxes, and other multi-sided shapes. A miter angle calculator finds this angle from the number of sides in the finished shape.

For a regular polygon, a shape with equal sides and equal angles, each corner needs two matching angled cuts, one on each board. A square picture frame needs a 45° cut on each board end. A hexagonal planter box needs a 30° cut. The angle depends only on how many sides the finished shape has.

Miter Angle Formula

The miter angle for a regular polygon is:

Miter angle = 180° ÷ n

Here, n stands for the number of sides in the polygon.

This comes from the geometry of the polygon. At each corner, the exterior angle (the angle you would turn through walking around the outside of the shape) equals 360° ÷ n. A miter joint splits that turn evenly between the two boards that meet there, so each board is cut at half of the exterior angle: 180° ÷ n.

The same result can be reached from the interior angle. The interior angle of a regular polygon with n sides is (n − 2) × 180° ÷ n. Subtracting this from 180° and dividing by two also gives 180° ÷ n.

How to Calculate a Miter Angle

1. Count the number of sides in the finished shape.
2. Divide 180 by that number.
3. Set the saw to the resulting angle.
4. Cut each piece at that angle on both ends, then join the pieces.

A valid regular polygon needs at least 3 sides. The calculator accepts whole numbers from 3 to 100. Below 3 sides there is no closed shape, and above 100 sides the corners become so shallow that individual cuts are no longer practical.

Worked example

Picture frame, 4 sides:

180° ÷ 4 = 45°

Each of the four frame pieces is cut at 45° on both ends. When joined, each corner forms a 90° angle, and the four corners close the square frame.

Planter box, 6 sides:

180° ÷ 6 = 30°

Each of the six boards is cut at 30° on both ends. The corners meet at 120°, the interior angle of a regular hexagon.

Gazebo roof edge, 8 sides:

180° ÷ 8 = 22.5°

Each of the eight pieces is cut at 22.5° on both ends, meeting at 135° corners to form a regular octagon.

Reading the Diagram

The calculator draws the polygon and shades the corner at its top vertex. That shaded wedge is the interior corner angle, (n − 2) × 180° ÷ n, which is the angle the two mitred board ends form once they are joined. It is not the saw setting. For a square frame the wedge reads 90°, while the saw is set to 45°, half of what the joint removes from the two ends together.

Common Miter Angles for Regular Polygons

Miter Cuts vs. Bevel Cuts

A miter cut angles across the width of a board, changing the direction it points when laid flat. A bevel cut angles through the thickness of a board, tilting the face of the cut. A compound cut combines both and is common in crown molding, where the trim sits at an angle between a sloped ceiling and a wall.

The angles in the table above are miter cuts only, for corners that lie flat. A compound cut is needed only when the joint itself is not flat.

Frequently asked questions

What is a miter angle?

It is the angle a saw is set to when cutting board ends that will meet at a corner. For a regular polygon with n sides, the miter angle equals 180° divided by n.

What miter angle do I use for a square frame?

45°. Cutting four boards at 45° on both ends and joining them forms the four 90° corners of a square frame.

What miter angle do I use for a hexagon?

30°. Six boards cut at 30° on both ends form a regular hexagon with 120° interior corners.

Does this formula work for irregular shapes?

No. The 180° ÷ n formula only applies to regular polygons, where every side and every angle is equal. An irregular shape needs each corner angle measured and calculated on its own.

Why does a 45° miter joint sometimes leave a gap?

A saw that is not calibrated exactly to 45°, a board that shifted during the cut, or wood that has swelled or shrunk from moisture can all leave a small gap. Checking the saw's calibration and cutting fresh, tight-fitting pieces usually solves it.

What is the difference between a miter cut and a bevel cut?

A miter cut angles across a board's width. A bevel cut angles through a board's thickness. Crown molding often needs both cuts at once, known as a compound cut.