Sheet metal
Miter cut calculator
In short
For two sections of the same width joining at the angle φ, the joint follows the bisector and each piece is cut at θ = 90° − φ / 2, measured from the square cut. With different widths l1 and l2, β = atan( l1 sin φ / (l2 + l1 cos φ) ), then θ1 = 90° − β and θ2 = 90° − (φ − β).
Calculation
Example calculated with the tool’s default values.
Inputs
- Section 1 width
- 60 mm
- Angle between the two sections
- 90°
Results
- Cut angle of each piece
- 45.00°From the square cut
Worked example
Every value in this section belongs to the example, calculated with the tool’s default values.
Other results of the example
- Cut angle, piece 2
- 45.00°From the square cut
- Joint / axis of piece 1
- 45.00°
- Joint / axis of piece 2
- 45.00°
- Joint length (cut length)
- 84.9 mm
- Outer → inner point, piece 1
- 60.0 mmOffset measured along the piece
- Outer → inner point, piece 2
- 60.0 mm
- Sharp-edge geometry on the width seen in the plane of the bend: saw kerf, weld gap and tube wall thickness are not deducted. Indicative result.
Step-by-step calculation
Equal widths: the joint follows the bisector
β = φ / 2 ; θ = 90° − φ / 2
β = 90.00 / 2 = 45.00° ; θ = 90 − 45.00 = 45.00°
Joint length
|OI| = l1 / sin β
60.0 / sin 45.000° = 84.85 mm
Marking offsets
d1 = l1 / tan β1 ; d2 = l2 / tan β2
d1 = 60.00 mm; d2 = 60.00 mm (from outer tip to inner tip, along the part)
How to use it
- Enter the width of section 1, seen in the plane of the bend.
- Enter the width of section 2, or leave it empty for two identical sections.
- Enter the angle between the two sections, from 5° to 175° (90° = square corner).
- Read the cut angle of each piece (of piece 1 if the widths differ).
Frequently asked questions
What is the miter cut angle formula?
φ is the inside angle of the corner, between the two axes. With two sections of the same width, the joint follows the bisector: β = φ / 2 and each piece is cut at θ = 90° − φ / 2, from the square cut. In other words, the cut angle of each piece is (180° − φ) / 2.
How do I cut a 45° miter?
For a square frame corner (φ = 90°) with identical sections: β = 90.00 / 2 = 45.00° and θ = 90 − 45.00 = 45.00°. With a 60 mm width, the joint is 60.0 / sin 45.000° = 84.85 mm long, and the inner point is 60.00 mm from the outer point, along the piece.
How do I miter two sections of different widths?
The joint runs from the outer corner to the inner corner I = ((l2 + l1 cos φ) / sin φ ; l1). Its angle with piece 1 is β = atan( l1 sin φ / (l2 + l1 cos φ) ), then θ1 = 90° − β and θ2 = 90° − (φ − β), with θ1 + θ2 = 180° − φ. The two pieces are not cut at the same saw setting.
How do I find the miter cut length?
The joint length, from the outer point to the inner point, is |OI| = l1 / sin β. For marking, the offset between the two points along each piece is d1 = l1 / tan β1 and d2 = l2 / tan β2.
Limits
Indicative results: sharp-edge geometry on the width seen in the plane of the bend. Saw kerf, weld gap and tube wall thickness are not deducted. A negative cut angle means the point flips to the other side: mark it out before cutting. Below 30°, the point is very thin.
Sources
- Coupe d’onglet : bissectrice pour deux largeurs égales, θ = 90° − φ/2 : Géométrie plane ; angle de coupe = (180° − φ) / 2 pour chaque pièce
- Largeurs différentes : angle du joint avec chaque pièce, arctan(a / b) à 90° : Omni Calculator, « Miter angle calculator » ; formule générale ci-dessus déduite de la géométrie des bords intérieurs (link)