Angle Distribution For Non-90º Miters
In any craft, it is common to have to make a miter cut on a piece of trim, siding, beam, joist, or basically anything. The most common miter angle is a 45º angle which when combined with another 45º angle forms a 90º corner. In this post I will explain a few simple miter angle cases and will then explain how to calculate the angle distribution for non-90º miters of pieces that have different thicknesses.
Case 1: A 90º Mitered Corner Where the Width of the Mitered Pieces is Constant
Think of the casing trim around the perimeter of a doorway. At either of the upper corners, a 90º corner is formed by the 2 pieces of trim that meet at that corner. If the 2 pieces of trim have the same width, then they have a 45º angle cut on them to form the 90º corner. Below is a hand sketch of the intersection of two identical flat trim boards that intersect at 90º. It is quite simple to make this miter cut. You simply set your miter saw to 45º and make this cut. If you are cutting by hand, you can construct the 45º miter line with a combination square on the work piece and cut along the 45º angled line. The key with miters is to ensure that the length of the miter line is the same on the two pieces being mitered together. This condition is key to developing a formula for the calculation of more complicated miter cuts.
Case 2: A 90º Mitered Corner Where the Width of the Mitered Pieces is Different
Now consider the same casing trim on the same doorway, but now the 2 intersecting trim pieces have different widths. This is a common condition as well and it too is simple to layout the miter cut. Now the angular distribution is different for each piece of trim. If cutting by hand, the actual angle of each cut does not matter, you simply construct the miter line by making an initial mark on the outside of the trim, making a reference mark on the inside surface perpendicularly aligned with the outside mark, measuring over a distance that is equal to the width of the adjoining piece of trim, and connecting these two points.
If you wish to cut these pieces of trim in a miter saw, you will want to know the angles to set the miter saw to for each piece. These can be be calculated using the formulae below:
Case 3: A non-90º Mitered Corner Where the Width of the Mitered Pieces is Different
Now let's consider 2 intersecting pieces of trim that not meet at a non-90º angle, but they also have different widths. The hand sketch below shows this condition.
In some cases, it is feasible to lay two pieces of trim across one another at the desired angle and mark the intersection points at the inside and outside surfaces. Connecting these two intersection points creates the miter line where the 2 pieces meet. However this method is not always feasible. If needed, you can calculate the miter angle for each piece with the following formula.
For those interested, here is my derivation of this formula:
I typically use this equation when installing metal coping on commercial roofs. It would be dangerous to lay loose pieces of metal on the edge of a roof to mark out the intersection points. A brief gust of wind could send a large piece of sheet metal sailing down to the ground potentially causing injury or death to a bystander. This formula allows you to safely cut the piece to fit perfectly without this risk.
- Jake Townsend