Hi Mike.

Due to similar triangles, corresponding side length ratios are equal:

Thus, x + 13 ¹/₄ = 22 ¹/₄, which you had already calculated. Knowing x, we can now calculate the really useful radii r and R by using Pythagorus:

So now we know we are cutting from an annulus with inside radius 14 ¹⁵/₁₆" and outside radius 37".

Now we need to consider the portion we must cut.

We know the outside circumference we want remaining is the circumference of the larger radius of the truncated cone. That circumference is the circumference of a circle with radius 29 ⁹/₁₆":

whereas the circumference of the outside of the annulus pattern is

The ratios of these circumferences is the same as the ratio of the angles through which they turn. That is, since the full circle of the annulus is 360 degrees, we can calculate what the angle θ is by using the common ratios:

When we solve for θ we get:

and now substitute in our earlier expressions for c and C to find the value of θ

Now we have our pattern: It is a portion of an annulus whose outside radius is 37, inside radius is 14 15/16, missing sector is 72.4°.   Cheers,    Stephen La Rocque