Miter Angle Calculator
Find the two board cut angles and practical miter-saw settings for equal or unequal board widths.
Joint and board details
Output values
Cut setup summary
| Board | Width | Miter angle | Miter-saw setting | Role |
|---|---|---|---|---|
| Board A | 10 cm | 40.89° | 49.11° | Narrower board |
| Board B | 15 cm | 79.11° | 10.89° | Wider board |
How to use this miter angle calculator
What this calculator does
This calculator determines the two cut-line angles needed when the ends of two boards meet at a chosen joint angle. It handles both equal-width boards and unequal-width boards, where simply dividing the joint angle in half would be wrong. It also reports a practical complementary miter-saw setting for each board, calculated as the absolute difference between 90° and the board's miter angle. The result is a geometric setup estimate: it does not account for blade kerf, spring-back, warped stock, bevel cuts, compound crown-molding geometry, or a particular saw manufacturer's scale convention.
When to use it
Use it when laying out picture frames with non-square corners, joining trim boards of different face widths, planning polygonal frames, or checking an angled casing or molding transition before making a test cut. It is especially useful when two boards have unequal widths because the two miter angles will usually differ even though their sum still equals the chosen joint angle.
How to calculate
- The calculator opens with a ready-to-use demonstration: a 120° joint, board A at 10 cm, and board B at 15 cm. The first results and a validated Excel workbook are available immediately.
- Replace Joint angle (θ) with the inside angle you want the finished boards to form. Enter a decimal number greater than 0 and less than 180.
- Enter Width of board A and Width of board B. Choose millimeters, centimeters, or inches. Changing the unit converts both current widths so the physical size does not change.
- Read Board A miter angle (α) and Board B miter angle (β). Their sum should equal the joint angle. Then check each saw setting, which is the complementary angle often shown on a conventional miter-saw scale.
- Use Download Excel to save the current inputs, outputs, formulas, and cut summary. Reset clears the demonstration and calculated state; Excel export stays disabled until all required values are complete and valid again.
Input guide
Joint angle (θ) is required and is entered in degrees using plain decimal notation, such as 120 or 90.5. Values at or below 0° and at or above 180° are rejected because they do not define the intended open two-board joint. Increasing θ increases the combined amount of miter angle shared by the two boards. A common mistake is entering the saw's scale reading instead of the desired finished inside angle.
Width of board A and Width of board B are required positive measurements across the board faces. Enter decimals such as 10, 15.5, or 3.25 and use one shared unit: mm, cm, or in. Larger width on one board shifts more of the joint angle to that wider board under the adopted geometry. Do not mix units between boards or enter board length instead of face width. The unit selector is a control, not a formula change: it converts both widths while preserving their ratio.
Output guide
Miter angles α and β is the primary pair of geometric cut angles in degrees. Board A miter angle (α) and Board B miter angle (β) show the individual values. Their exact identity is α + β = θ. When the widths are equal, both outputs equal θ ÷ 2. With unequal widths, the narrower board generally receives the smaller miter angle. Board A saw setting and Board B saw setting are complementary values, |90° – miter angle|, and should be interpreted against your saw manual. The summary pills repeat the current joint, board widths, and angle-sum cross-check. The Cut setup summary table repeats each board's width, miter angle, saw setting, and width role for shop use.
Worked example
For the startup example, θ = 120°, board A = 10 cm, and board B = 15 cm. Board A is narrower, so the calculator uses α = arctan[(10 × sin 120°) ÷ (15 + 10 × cos 120°)]. This gives α = 40.89339465°, displayed as 40.89°. Board B receives the remaining angle: β = 120° – 40.89339465° = 79.10660535°, displayed as 79.11°. The saw settings are |90° – 40.89339465°| = 49.11° and |90° – 79.10660535°| = 10.89°. The two displayed miter angles sum to 120.00° apart from normal rounding.
Learn more
The unequal-width formula uses inverse tangent; OpenStax explains the arctangent and other inverse trigonometric functions. For unit context, NIST describes the radian as the coherent SI unit of plane angle, while this shop calculator displays the more familiar degree. Before operating a saw, review OSHA's miter-saw guarding and safety guidance.
How the geometry works
For equal board widths, symmetry makes the two cuts identical, so each miter angle is exactly one-half of the joint angle. For unequal widths, the cut line must cross each board at a different orientation while still meeting at one common corner. Let n be the narrower width, m the wider width, and θ the finished joint angle:
narrower angle = atan2(n × sin θ, m + n × cos θ)
wider angle = θ – narrower angle
The calculator uses the two-argument arctangent form, atan2, because it preserves the correct quadrant when the denominator becomes negative for very open joints. Widths may be in any common unit because only their ratio affects the angles. Converting from centimeters to inches therefore changes the displayed widths but not the calculated angles.
Practical cutting notes
Transfer calculated values carefully and make test cuts from scrap before cutting finished material. Real joints can differ from the theoretical result because of blade kerf, fence alignment, board twist, measurement error, and the actual wall or frame angle. Measure the real corner in more than one place when possible, keep reference edges clearly marked, and confirm which side of the blade preserves the finished dimension.