Angle for arched window casing

Methods for making the calculation. October 30, 2003

Question
Is there a way of determining what angle to make the cuts on the legs of the casing of an arched window where they meet the arched top?

Forum Responses
(Architectural Woodworking Forum)
From contributor A:
Technically, if you determine the angle of a line tangent to the very end of the eyebrow arched casing, you can then measure the angle between that and the vertical casing. Then you can divide that by two and get your cut angle.

In reality, I tell our customers to make an approximate visual 45 degree cut on the end of the eyebrow (we ship them overlong) and then make a corresponding cut on a short scrap of the straight moulding. Tack the curve in place and adjust the angle on the straight until it meets properly.

Once this is done, you can examine how cleanly the profile transitions the joint. If it doesn't line up properly, you adjust the angle on the eyebrow a bit and try again. One or two trial and error cuts and you should have it.

Finally, cut the angle on the final straight moulding and duplicate the eyebrow angle on the other end of the curve.

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From the original questioner:
Thanks for the help. Maybe I should post this question on a math forum. It was the technical solution I was looking for. I'm sure someone out there knows it. I have a bunch of these to do and I didn't want to do the trial and error method.

Here is my trial and error method… I take the legs and mark my final height (length) on the outside. I then place the legs parallel at the exact final width apart (important!). Then place the outside of the arch on the marks indicating the finished height. Make a mark on the inside of the legs where the inside of the arch intersects it. Then mark the arch at the points where it intersects the legs inside and out. Draw a straight line connecting those marks and you have your angle.

This works really well in theory, and if you are doing mouldings without any profile you can be very accurate because both the inside and outside of the mouldings are touching. In reality there is always some inside profile keeping you from lining the two up exactly. As I write this, I realize you can take a small square and butt it up to the point where the two mouldings intersect and that will give you a good way of transferring the same mark to both pieces. I'd still rather calculate it, though.

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From contributor B:
Yes, there is.
1 rad = 57.2958 degrees
1 rad is a length of arc equal to radius (6.28 rads in a circle).
So divide the distance of arc by the radius to get (no. of rads) in arc.
Multiply (no. of rads) with 57.2958 degrees to get angle of arc.
Divide angle in half to get the angle of miter (as determined by a plumb line).
Subtract miter angle from 90 to get cut angle for saw.
By the way, do you know the radius? Can you measure the arc?

This being said... In the real world I draw two parallel lines on the wall representing the width of the casing for the legs. Lines are extended up past any intersecting points of the arch casing. I put the arch in place temporarily (temporarily = "John get your butt over here and hold this damn thing in place"). I transfer the intersecting marks to inside and outside edges of the arched casing. Also transfer intersecting marks to the wall to give me an angle to copy for the miter cut to legs.