Angle Cut Calculator

Angle cutting is the process of cutting boards at a non-perpendicular 90-degree angle so that two boards can cleanly form a corner. This calculator will calculate the angles required for three common applications: mitering to join two boards in a corner, regular polygonal frames and diagonal braces set on right angles.

Select a pattern, enter the measurements and read off the required marking angle or settings to be pre-set on your miter saw. Each result is based on the same trigonometric functions used in manual calculations for woodworking but performed faster and without rounding errors.

What is a miter?

A miter is a diagonal cut along the width of an object so that when two objects are joined together they do not have rough edges. By mitering both objects and joining them, you create a corner. A typical example would be picture frames; four pieces, four corners, each end being mitered at 45 degrees.

The miter angle is not the same as the saw setting. The miter angle is measured from the edge of the workpiece while the saw setting is measured vertically, so a miter saw will show zero when cutting straight across. The sum of both values always equals 90 degrees, so a miter angle of 45 degrees corresponds to a saw setting of 45 degrees, while a miter angle of 30 degrees corresponds to a saw setting of 60 degrees. This calculator shows both values.

How do you calculate the miter angle?

Two boards must form a corner so the two cut surfaces of both boards must bridge the entire angle between them. Each case is based on this simple concept.

Plates of equal width

If two boards are the same width they will share that angle evenly so each board's cut angle is half of the total.

A 90-degree angle will result in each tile being cut at a 45-degree angle. A 120-degree angle (a common angle for hexagons) will result in each tile being cut at a 60-degree angle. This is the only situation that most decorative work requires.

Planks of different widths

If the plates are of different widths, then the wider plate will take up more of the angle. The cut angles therefore won't be equal but they still add up to the total angle. If one plate is w wide and another is W wide meeting at an angle of J degrees,

As a check two boards with widths of 60 mm and 80 mm respectively are joined at a right angle of 90 degrees. The narrower board must be cut by 36.87 degrees and the wider board by 53.13 degrees, which adds up to 90 degrees. At an angle of 120 degrees the same two boards need to be cut by 46.10 and 73.90 degrees respectively. If the board widths are equal this formula automatically reduces to half the angle.

Cutting angle for polygonal frames:

A regular framework of equally wide boards such as a hexagonal planter or octagonal mirror simply repeats the application of this formula at each corner. For a frame with N sides, the interior angle is (N minus 2) multiplied by 180, divided by N; any board meeting at a corner must be cut to half that angle.

Six plates each cut at 60 degrees can form a hexagon, while eight plates each cut at 67.5 degrees can form an octagon. When you enter the number of sides in polygon mode, the tool will calculate the result for you automatically.

Cutting angle for diagonal supports:

A short diagonal brace, also called a knee, is used to strengthen corners between posts and beams or walls and floors. Both ends of it must be cut accordingly so that each end fits flat against the respective member. The brace, post, and beam form a right triangle: the vertical side along the post is A, the horizontal side along the beam is B, and the hypotenuse C is the outside edge of the brace.

Angle alpha is the bevel angle at where the bearer meets the stud and beta is the bevel angle at where the bearer meets the cross beam. Due to the actual thickness t of the board each beveled edge shifts the inside edge compared to the outside edge: The top edge is shifted back by a distance of t divided by tan(alpha) and the bottom edge is shifted back by a distance of t divided by tan(beta). Assuming that the bearer spans an angle with A being 40 cm, hypotenuse C being 60 cm. Then B comes out to be 44.72 cm, alpha is 48.19 degrees, beta is 41.81 degrees and a board of 10 cm width will loose 8.94 cm on the top side and 11.18 cm on the bottom side.

Enter any two of the three values for the legs and hypotenuse, leaving the third blank; the calculator will compute the third value using the Pythagorean Theorem, then give both rake angles as well as finished interior and exterior lengths.

How to use this angle calculator:

First select the mode. The Miter Joint is used to connect two boards in a corner, the Polygon Frame for regular polygon frames and Diagonal Bracing for rectangular supports. Each mode shows only the required fields.

Enter the measurements in any units. The length can be switched between millimeters, centimeters, meters, inches and feet, and the angle can be switched between degrees, radians and percent grade. The results will update synchronously with your inputs, and a summary card will summarize the cut requirements into one sentence that you can take to the saw. Open the step-by-step panel to see how the calculations are done, and open the charts to compare different angles at a glance.

What tools can be used to cut angles?

A miter saw is often the best choice: a circular blade mounted on an arm that pivots along a fence to a preset angle. Most have stops for 15, 22.5, 30 and 45 degrees that snap into place; their scales also show all settings between those angles. Table saws with miter gauges, combination miter saws, jigsaws, circular saws or simple hand saws used in a miter track can cut angles as well, but power tools are much more accurate at holding the angle than manual methods.

Regardless of the tool used, a test run on scrap material should always be carried out before the actual cut. Even a deviation of one degree in miter cutting leads to visible gaps at the corners; adjusting the angle for a test step is much faster than having to re-cut the entire component.

Miter cut versus bevel cut

A miter cut is a cut where the saw blade moves across the surface of an object and changes angle as viewed from above. A bevel cut, on the other hand, tilts the saw blade so that the cut runs along the thickness of the object rather than its width. Bevel cuts are used to create chamfered edges, and can be combined with miter cuts to form complex angles in molding. This calculator deals with flat miters; a bevel cut is another type of tilt on the saw blade.

Symbols used

This calculator is a general woodworking and problem solving tool. It assumes the material is flat and straight and that the cutting faces are square. In reality, material defects, saw kerfs, and uneven walls mean it's always worth doing a test run on scrap material.